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Network Theory in Finance

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■ Relation between regional uncertainty spillovers in the global banking system Sachapon Tungsong, Fabio Caccioli and Tomaso Aste

■ The quest for living beta: investigating the implications of shareholder networks Matthew Oldham

■ A stock-flow consistent macroeconomic model with heterogeneous agents: the master equation approach Matheus R. Grasselli and Patrick X. Li

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The Journal of Network Theory in FinanceEDITORIAL BOARD

Editor-in-ChiefTiziana Di Matteo King’s College London

Associate EditorsFranklin Allen Brevan Howard Centre

& Imperial College Business SchoolIgnazio Angeloni European Central BankTomaso Aste University College LondonStefano Battiston University of ZurichChristian T. Brownlees Pompeu Fabra

UniversityGuido Caldarelli IMT LuccaAndrew Coburn RMS, Inc.

& University of CambridgeRama Cont Imperial College London

& CNRS (France)Ben Craig Deutsche BundesbankRodney Garratt Federal Reserve Bank

of New YorkCo-Pierre Georg Deutsche BundesbankAndrew G. Haldane Bank of EnglandSergey Ivliev Perm State UniversityDror Kenett Johns Hopkins

University

Iman Van Lelyveld De NederlandscheBank

Thomas Lux University of Kiel& University Jaume I

Rosario Nunzio Mantegna CentralEuropean University & Palermo University

Serafín Martínez Jaramillo Banco deMéxico

Camelia Minoiu International MonetaryFund

Yaacov Mutnikas Markit GroupPeter Sarlin Hanken School of

EconomicsKimmo Soramäki Financial Network

Analytics Ltd.Didier Sornette ETH ZurichMurat Unal SONEAN GmbH

& Funds at WorkWei-Xing Zhou East China University of

Science and Technology

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The Journal of

Network Theoryin Finance

The journalFinancial institutions and markets are highly interconnected, but only recently hasliterature begun to emerge that maps these interconnections and assesses their impacton financial risks and returns. The Journal of Network Theory in Finance is an interdis-ciplinary journal publishing academically rigorous and practitioner-focused researchon the application of network theory in finance and related fields. The journal bringstogether research carried out in disparate areas within academia and other researchinstitutions by policymakers and industry practitioners.

The Journal of Network Theory in Finance publishes data-driven or theoreticalwork in – but not limited to – the following areas.

� Empirical network analysis that enables better understanding of financialflows, trade flows, input–output tables, financial exposures or market inter-dependencies.

� Modeling and simulation techniques for measuring interdependent financialrisks.

� New metrics and techniques for identifying central, vulnerable or systemicallyimportant institutions and markets in financial networks.

� Network modeling of time-series data for financial risk management, assetallocation and portfolio management.

� Social network analysis (SNA) in finance, such as using social network datafor making credit and investment decisions.

� Applied network visualization techniques that improve the communication offinancial risks and rewards.

� Analysis of counterparties and their risk exposure from interconnectivity withthe financial system and regulatory strategies for improving financial stability.




The Journal of Network Theory in Finance Volume 4/Number 2

CONTENTS

Letter from the Editor-in-Chief vii

RESEARCH PAPERSRelation between regional uncertainty spillovers in the global bankingsystem 1Sachapon Tungsong, Fabio Caccioli and Tomaso Aste

The quest for living beta: investigating the implications of shareholdernetworks 25Matthew Oldham

A stock-flow consistent macroeconomic model with heterogeneousagents: the master equation approach 47Matheus R. Grasselli and Patrick X. Li

Editor-in-Chief: Tiziana Di Matteo Subscription Sales Manager: Aaraa JavedPublisher: Nick Carver Global Key Account Sales Director: Michelle GodwinJournals Manager: Sarah Campbell Composition and copyediting: T&T Productions LtdEditorial Assistant: Ciara Smith Printed in UK by Printondemand-Worldwide

© Infopro Digital Risk (IP) Limited, 2018. All rights reserved. No parts of this publication may be reproduced,stored in or introduced into any retrieval system, or transmitted, in any form or by any means, electronic,mechanical, photocopying, recording or otherwise without the prior written permission of the copyright owners.

Composition and copyediting: T&T Productions LtdPrinted in UK by Printondemand-Worldwide




LETTER FROM THE EDITOR-IN-CHIEF

Tiziana Di MatteoKing’s College London

Welcome to the second issue of Volume 4 of The Journal of Network Theory inFinance.

This issue starts with “Relation between regional uncertainty spillovers in the globalbanking system” by Sachapon Tungsong, Fabio Caccioli and Tomaso Aste, a paperthat focuses on the quantification of systemic risk from market data. It is inspiredby a 2009 work by Francis Diebold and Kamil Yilmaz, which proposed a methodbased on forecast error variance decomposition to estimate, using market data, net-works of interdependencies between firms, and used the connectedness of the esti-mated networks to quantify spillovers of uncertainty between variables. The presentpaper improves on that work, representing a step forward in our understanding of theconnectedness of banking systems. In particular, this paper generalizes Diebold andYilmaz’s methodology to an exponentially weighted daily returns and ridge regular-ization on vector autoregression and forecast error variance decomposition. Moreover,it estimates the time evolution of connectedness in the following three regional bank-ing systems: North America, Southeast Asia and the European Union. This allows theauthors to perform a comparative analysis of the three regions and a quantificationof the existence of causal relations between different regions. Aside from the newresults and robust analysis this paper reports, I very much enjoyed reading its verywell-crafted and exhaustive literature review. I am sure our readers will enjoy it too.

Our second paper, “The quest for living beta: investigating the implications ofshareholder networks” by Matthew Oldham, looks at financial markets as complexsystems and applies network theory to study their behavior. A vast literature hasdeveloped around this idea, and, in my opinion, the econophysics community hasadded significantly to its development. This paper contributes nicely to these studiesby analyzing the dynamical evolution of bipartite networks as well as the subsequentstock-by-stock and investor-by-investor networks formed for each quarter between2007 and 2010, comprised of the stocks in the Standard & Poor’s 500 (S&P 500) andthe US institutional investment managers that held them. It presents an interdisci-plinary and alternative type of study to the traditional standard mainstream economicand finance type of analysis, capturing a novel result: namely, a parallel movementin the density of the investment network and the volatility and value of the S&P 500index. Nowadays, it is always important to tackle problems by looking at data andusing approaches that are interdisciplinary by nature as well as new and able to beapplied to different systems.

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vii




“A stock-flow consistent macroeconomic model with heterogeneous agents: themaster equation approach” by Matheus R. Grasselli and Patrick X. Li, our third andfinal paper, is another example that proposes the alternative approach of agent-basedmodeling. It addresses a key theme in macroeconomics: the distinction between theactions of individual agents and aggregate behavior. This is an alternative to boththe aggregate-level Keynesian model and the representative-agent-based dynamicstochastic general equilibrium (DSGE) model, and it considers agents not constrainedby utility-maximizing behavior and aggregation not achieved through equilibrium.The authors’ main proposal is a mean-field-type approximation to a stock-flow con-sistent agent-based model with two types of firms and two types of households.This allows them to investigate the behavior of aggregate variables with respect toparameters that are difficult to estimate outside the model, such as the fraction ofexternal financing that firms raise by issuing new debt as opposed to equity. Agent-based modeling relies on numerical simulations, making it both computationally time-consuming and difficult to interpret; however, this paper shows how approximatingby means of a mean-field approach might provide results that are not achievableotherwise.

Let me end this editorial letter by thanking this issue’s authors for their valuable con-tributions, and by reminding you that Risk Journals is proud to sponsor NetSci 2018,the flagship conference of the Network Science Society, which aims to bring togetherleading researchers and practitioners working in the emerging area of network science.The conference fosters interdisciplinary communication and collaboration in networkscience research across computer and information sciences, physics, mathematics,statistics, the life sciences, neuroscience, environmental sciences, social sciences,finance and business, arts and design (see www.netsci2018.com).

Journal of Network Theory in Finance 4(2)




Journal of Network Theory in Finance 4(2), 1–23DOI: 10.21314/JNTF.2018.040

Research Paper

Relation between regional uncertaintyspillovers in the global banking system

Sachapon Tungsong,1,2 Fabio Caccioli1,2,3 andTomaso Aste1,2

1Department of Computer Science, University College London, Gower Street, London WC1E 6BT,United Kingdom; emails: [email protected], [email protected], [email protected] Risk Centre, London School of Economics and Political Sciences, Houghton Street,London WC2A 2AE, United Kingdom3London Mathematical Laboratory, 8 Margravine Gardens, London W6 8RH, United Kingdom

(Received October 25, 2017; revised March 2, 2018; accepted April 20, 2018)

ABSTRACT

We report on time-varying network connectedness within three banking systems:North America (NA), the European Union (EU) and Southeast Asia (ASEAN).Diebold and Yilmaz’s original method is improved by using exponentially weighteddaily returns as well as ridge regularization on vector autoregression (VAR) and fore-cast error variance decomposition (FEVD). We compute the total network connect-edness for each of the three banking systems, which quantifies regional uncertainty.Results over rolling windows of 300 days during the period from January 2005 toOctober 2015 reveal changing uncertainty patterns that are similar across regions,demonstrating common peaks associated with identifiable exogenous events. Lead–lag relationships among changes of total network connectedness of the three systems,quantified by transfer entropy, reveal that uncertainties in the three regional systemsare significantly causally related, with the NA system having the largest influence onthe EU and ASEAN.

Keywords: systemic risk; forecast error variance decomposition (FEVD); connectedness; spillovereffects; banking networks.

Corresponding author: F. Caccioli Print ISSN 2055-7795 j Online ISSN 2055-7809© 2018 Infopro Digital Risk (IP) Limited

1 Journal of Network Theory in Finance





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2 S. Tungsong et al

1 INTRODUCTION

Financial markets are increasingly becoming more interconnected (Moghadam andVinals 2010), and shocks initially affecting one part of the system can quickly prop-agate to the rest of it. Therefore, understanding the patterns of distress propagationwithin financial markets is important to characterize systemic risk. After the globalfinancial crisis of 2007–9, significant effort has been devoted to understanding themechanics of distress propagation within banking systems. One strand of the litera-ture has focused on modeling the processes through which contagion may occur ininterbank networks (see, for example, Glasserman and Young (2016) and Caccioliet al (2018) for recent reviews). Another strand of the literature has focused on thequantification of systemic risk from market data (see Adrian and Brunnermeier 2016;Brownlees and Engle 2016). In particular, Diebold and Yilmaz (2009) proposed amethod based on forecast error variance decomposition (FEVD) of estimating frommarket data networks of interdependencies between firms, and they used the con-nectedness of the estimated networks to quantify spillovers of uncertainty betweenvariables.

In this paper, we use the methodology of the aforementioned work by Dieboldand Yilmaz (2009) to estimate the time evolution of connectedness in three regionalbanking systems: North America (NA), the European Union (EU) and SoutheastAsia (ASEAN). Through VAR and FEVD, we compute the pairwise connectednessbetween pairs of banks in each region, and we aggregate such pairwise connectednessto compute a measure of total connectedness for the region.

The time-varying total connectedness computed for each banking system, froma 300-day rolling window during the period from January 2005 to October 2015,indicates temporal changes of systemic risk, with peaks during major crisis eventsand troughs during normal periods. Analogous results have been observed in otherfinancial systems and different regions (Alter and Beyer 2014; Chau and Deesomsak2014; Demirer et al 2015; Diebold and Yilmaz 2009, 2012, 2014; Fengler and Gisler2015). It has to be stressed that, unlike Diebold and Yilmaz (2009), who view allfinancial institutions as belonging to one global system, here we group banks into threeregional banking systems. In this way, we can perform a comparative analysis betweenthe different regions, which allows us to highlight similarities and differences betweenthem. Further, this allow us to quantify the existence of causal relations betweendifferent regions. Note that combining all the banks together could be somehowmisleading, because the banks’equities in the three banking systems trade in differentstock markets that have significantly different trading hours.

The main results of our analysis are as follows. First, we note that the structureof the peaks in the three regional banking systems is very similar, with large peaksassociated with significant, identifiable major events. Although the overall patterns

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Relation between regional uncertainty spillovers in the global banking system 3

are similar, we observe two important differences between the systems. The first is thefact that the overall scale of connectedness is different, with the NA banking systembeing more interconnected than that in the EU, which, in turn, is more interconnectedthan the ASEAN system. Second, we uncover the existence of lead–lagged relationsbetween the different time series. To quantify this effect, we compute the transferentropy between the time series associated with changes of connectedness in thedifferent regions, and we uncover the existence of significant net information flowsfrom NA to the EU, from NA to ASEAN, and from the EU to ASEAN. The robustnessof our finding is tested using different measures for transfer entropy. In particular,we find consistent results for the net information flow both with a linear measureof transfer entropy (which corresponds to a Granger causality analysis) and withnonlinear measures of different parameters. We also retrieve similar causal relationsfor both one-day and five-day returns. To the best of our knowledge, this causalitystudy between regional uncertainties is the first of its kind.

The rest of this paper is organized as follows. In Section 2, we present a litera-ture review and place our paper in the context of previous works. In Section 3, wedescribe the used data, while Section 4 provides a brief description of our methodol-ogy. Section 5 illustrates and discusses the main results of the paper. We present ourconclusions in Section 6.

2 LITERATURE REVIEW

The literature on systemic risk and contagion in the banking network can be broadlyclassified into two categories. The first category comprises network models that aimto describe various causal mechanics of financial contagion, which can be calibratedwith balance-sheet data (Birch and Aste 2014; Cont et al 2010; Degryse and Nguyen2007; Furfine 2003; Müller 2006; Upper 2011; Upper and Worms 2004). The sec-ond category comprises econometric models that aim to identify spillover effectsexclusively from market data, without making assumptions about the dynamics ofdistress propagation between banks (Adrian and Brunnermeier 2016; Brownlees andEngle 2016). Our paper is close to this second strand of the literature, as we try tounderstand whether market data carries information about the level of interconnected-ness between banks and how exogenous shocks can be amplified by the endogenousdynamics of financial markets.

Network models of contagion go back to the seminal work of Allen and Gale(2000), who showed how the stability of the banking system is affected at equilib-rium by the pattern of interconnections between banks, and to the work of Eisenbergand Noe (2001), who demonstrated how to consistently compute a clearing vectorof payments in a network of interbank claims. The relation between the structure ofan interbank network and its stability has also been extensively explored within the

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4 S. Tungsong et al

context of nonequilibrium network models (see, for example, Bardoscia et al 2015,2017; Battiston et al 2012; Cont et al 2010; Fricke and Lux 2015; Furfine 2003; Gaiand Kapadia 2010; Iori et al 2006; Kobayashi and Hasui 2014; Lenzu and Tedeschi2012; Nier et al 2007; Tedeschi et al 2012; Upper 2011), showing in particular theexistence of a tension between individual risk and systemic risk: what makes a bankindividually less risky might in fact increase the risk of a systemic failure (Beale et al2011). More recently, these analyses have been extended beyond interbank lendingnetworks to the study of networks of overlapping portfolios (Caccioli et al 2014; Corsiet al 2016; Huang et al 2013).Although these models have been insightful with regardto understanding the dynamics of financial contagion, and in some cases they havebeen applied to real data (see Upper (2011) for a review of the existing literature),there are clear challenges to their applicability. First, there is a lack of reliable data onbanks’ balance sheets, which makes it hard to calibrate models.1 Second, to obtain areliable assessment of systemic risk one has to capture all relevant types of intercon-nections between banks, as the interaction between different contagion channels cansignificantly change the stability of the system (Caccioli et al 2015).

Here, we take the complementary approach of inferring interdependencies betweenbanks from market data, which belongs to the second strand of literature mentionedabove. The advantage of this approach with respect to network modeling is that marketdata is readily available, and different types of interconnections between banks havealready been aggregated by the market. The drawback is that this approach does notprovide an explanation of how stress propagates between banks, and it relies on theunderlying assumption of market efficiency, which is not realistic (Shiller 2003). Nev-ertheless, one can assume that, although markets are not efficient, prices do reflect tosome extent the aggregate information (or expectations) about the underlying assets.There have been several contributions to this strand of the literature. In particular,Dungey et al (2005) provide a summary of empirical models of contagion up to 2005.More recent empirical work includes Diebold and Yilmaz (2009, 2012, 2014), Cac-eres et al (2010), Billio et al (2012), Claeys and Vasicek (2014), Lucas et al (2014),Musmeci et al (2015) and Brownlees and Engle (2016). Of particular relevance forour paper is the work of Diebold and Yilmaz (2009, 2012, 2014), which influencedsubsequent studies such as McMillan and Speight (2010), Bubák et al (2011), Fuji-wara and Takahashi (2012), Klößner and Wagner (2014), Alter and Beyer (2014),Chau and Deesomsak (2014), Demirer et al (2015) and Fengler and Gisler (2015).

1 Admati et al (2013) report that banks tend to find ways to get around regulations in order toinvest in mortgage-backed securities and derivatives via structured-investment vehicles, which areoff-balance-sheet items. Such leeway being allowed by regulations creates regulatory boundaries,making it difficult for outsiders to know what banks actually report.






This strand of contributions uses vector autoregression (VAR) and forecast error vari-ance decomposition (FEVD) to quantify the unpredictability of each variable in thenetwork. By using the VAR and FEVD methods, it is possible to disentangle the con-tribution to unpredictability due to endogenous interdependencies from that due toexogenous shocks. Following Diebold and Yilmaz, we will refer to this endogenouscomponent in our paper as total network connectedness, which therefore quantifiesthe transmission of shocks from banks within the system.

3 DATA

We collect daily stock prices from January 2005 to October 2015 of banks headquar-tered in NA (including the United States and Canada), the EU and ASEAN from theCompustat database. We select only financial institutions in the subindustry “Banks”(ie, large banks operating at the national level and having a GICS code of 40101010)and compute log returns from the daily closing prices for each bank. With the afore-mentioned criteria, our sample includes ten publicly listed banks in NA, sixty-sixbanks in the EU and thirty-nine banks in ASEAN that survived through the periodJanuary 2005–October 2015.

While we could analyze rolling windows in which the number of banks in operationvaries from one window to the next, we find that being able to see the evolution ofthe systems’ total connectedness given a constant number of banks provides somebaseline insight into how the same set of banks reacted to different economic andfinancial episodes over time. That being said, research that saw all surviving banksbeing accounted for in respective rolling windows would be an interesting avenue toexplore. In such a case, the dimensions of the rolling windows would likely be muchlarger and estimation techniques such as sparsity modeling would be needed.

All banks in the NA banking system have their stocks traded in the NewYork StockExchange (NYSE), while the EU and ASEAN bank stocks mostly trade in their ownnational stock markets. Lists of banks in all three regions as well as their summarystatistics are given in Tables 1–3.

The data was analyzed over rolling windows of 300 days and over the full period.Harris (1985) recommends using a sample size such that n > 50 C k, where k is thenumber of predictors. For our study, the minimum number of observations for eachrolling window is thus 50C63 D 113.We experimented with window sizes of 250, 500and 750 days and obtained similar results in terms of the overall shape, including peaksand troughs, of total connectedness. We chose the window size of 300 days becauseit is a good compromise between obtaining results with a reasonable margin of errorand making sure we cover the period of interest (March 2006 to November 2015).A window size of 500 days would provide results with a lower margin of error butcover the period from January 2007 onward, while a window size of 250 days would





6 S. Tungsong et al

TABLE 1 List of banks headquartered in NA (Canada and the United States) that haveactively traded between 2005 and 2015.

Daily Dailymean volatility

Bank name Country return (%) (%)

Canadian Imperial Bank (CIBC) CAN 0.01 1.82Bank of Montreal (BMO) CAN 0.01 1.69Royal Bank of Canada (RBC) CAN 0.03 1.73Toronto Dominion Bank (TD) CAN 0.03 1.65Bank of Nova Scotia (BNS) CAN 0.01 1.72

Citigroup (CITI) USA �0.08 3.70Bank of America Corp (BAC) USA �0.04 3.51Wells Fargo & Co (WFC) USA 0.02 2.86JP Morgan Chase & Co (JPM) USA 0.02 2.64US Bancorp (USB) USA 0.01 2.32

provide results with a higher margin of error but cover the period from January 2006onward.

4 METHODOLOGY

4.1 Total connectedness

Following the approach introduced by Diebold and Yilmaz (2009, 2012, 2014), weuse a variance decomposition whereby the forecast error variance of a variable isdecomposed into contributions attributed to each variable in the system. The approachis based on the VAR model, introduced by Sims (1980) (see Stock and Watson (2001),Cochrane (2005), Lutkepohl (2006) and Tsay (2010) for discussions, reviews andapplications).

VAR estimates the value of a set of N variables yt;1; : : : ; yt;N at time t from a linearcombination of their values in the past by performing a multidimensional regression.By using the vectorial representation Yt D .yt;1; : : : ; yt;N /T and considering the t �1

lag only, the regression can be written as Yt D AYt�1 C"t , with A an N �N matrixof coefficients. By iterating this formula and expressing it in terms of an orthonormalbasis of residuals wi;t (with var.wi;twj;t / D ıi;j ) (Cochrane 2005), one can write

yi;t D1X

sD0

NXj D1

�ij;swj;t�s: (4.1)






TABLE 2 List of banks headquartered in ASEAN that have actively traded between 2005and 2015.

Market cap Average VolatilityBank Country (US$ billion) return (%) (%)

Bank Rakyat Indonesia IDN 20.43 0.07 2.56Bank Permata IDN 0.54 0.02 1.93Bank Danamon IDN 2.23 0.00 2.73Bank Maybank Indonesia IDN 0.79 0.00 2.67Bank Cimb Niaga IDN 1.07 0.02 2.51Panin Bank IDN 0.17 0.03 2.68Bank Negara Indonesia IDN 6.66 0.04 2.50Bank Central Asia IDN 23.21 0.08 2.06Bank Mandiri IDN 15.75 0.05 2.54

Public Bank MYS 16.15 0.04 0.90Malayan Banking MYS 18.70 0.00 1.23RHB Capital MYS 3.73 0.03 1.58AMMB Holdings MYS 3.04 0.01 1.51AFFIN Holdings MYS 0.97 0.01 1.65Alliance Financial Group MYS 1.15 0.01 1.52BIMB Holdings MYS 1.35 0.03 2.13CIMB Group Holdings MYS 7.92 0.02 1.54Hong Leong Bank MYS 6.17 0.03 1.14

Philippine National Bank PHL 1.20 0.03 2.39Bank of Philippine Islands PHL 6.97 0.03 1.79China Banking Corp PHL 1.36 0.04 1.39Metropolitan Bank and Trust PHL 4.67 0.05 2.12Security Bank Corp PHL 1.86 0.07 1.87Rizal Commercial Bank Corp PHL 0.94 0.03 2.19Union Bank PHL 1.22 0.05 1.77BDO Unibank PHL 7.33 0.05 2.04

United Overseas Bank SGP 19.62 0.01 1.49DBS Group Holdings SGP 25.23 0.01 1.49Oversea-Chinese Banking SGP 22.71 0.02 1.33

Krung Thai Bank THA 6.79 0.02 2.11Siam Commercial Bank THA 11.44 0.03 2.02Bangkok Bank THA 8.04 0.02 1.81Bank of Ayudhya THA 6.15 0.03 2.41Kasikornbank THA 10.94 0.04 1.97TMB Bank THA 3.12 �0.01 2.40Kiatnakin Bank THA 0.91 0.00 1.94Tisco Financial Group THA 0.96 0.02 2.11Thanachart Capital THA 14.3 0.03 2.13CIMB Thai Bank THA 0.76 �0.01 2.75





8 S. Tungsong et al

TABLE 3 List of banks headquartered in the EU that have actively traded between 2005and 2015. [Table continues on next page.]

Daily VolatilityBank Country return (%) (%)

Oberbank Ag AUT 0.02 0.38Erste Group Bk Ag AUT �0.01 2.95

KBC Group Nv BEL 0.00 3.50Dexia Sa BEL �0.21 7.76

Hellenic Bank CYP �0.08 3.08

Komercni Banka As CZE 0.01 2.10

IKB Deutsche Industriebank DEU �0.13 3.90Commerzbank DEU �0.08 3.09DVB Bank Ag DEU 0.03 1.38HSBC Trinkaus & Burkhardt DEU 0.00 1.73Comdirect Bank Ag DEU 0.02 1.83Deutsche Postbank Ag DEU 0.00 2.15

Danske Bank As DNK 0.01 2.11Jyske Bank DNK 0.02 1.94Nordea Invest Fjernosten DNK 0.01 1.43Sydbank As DNK 0.03 1.93

Banco Santander Sa ESP 0.00 2.16BBVA ESP �0.01 2.12Banco Popular Espanol ESP �0.07 2.30Bankinter ESP 0.01 2.28Banco De Sabadell Sa ESP �0.02 1.89

BNP Paribas FRA 0.00 2.56Natixis FRA �0.01 3.12Societe Generale Group FRA �0.02 2.86Credit Agricole Sa FRA �0.02 2.78CIC (Credit Industriel Comm) FRA 0.00 1.41

Barclays Plc GBR �0.03 3.23HSBC Hldgs Plc GBR �0.02 1.72Royal Bank of Scotland Group GBR �0.10 3.91Standard Chartered Plc GBR 0.00 2.44Lloyds Banking Group Plc GBR �0.05 3.37

Piraeus Bank Sa GRC �0.22 5.04Attica Bank Sa GRC �0.23 5.88Eurobank Ergasias Sa GRC �0.31 5.52National Bank of Greece GRC �0.20 4.81Alpha Bank Sa GRC �0.15 4.69






TABLE 3 Continued.

Daily VolatilityBank Country return (%) (%)

Zagrebacka Banka HRV 0.00 2.58Privredna Banka Zagreb Dd HRV 0.01 2.37

OTP Bank Plc HUN 0.00 2.63

Unicredit Spa ITA �0.05 2.90Credito Emiliano Spa ITA 0.00 2.26Intesa Sanpaolo Spa ITA 0.00 2.61Banca Popolare Di Sondrio ITA �0.01 1.83Banca Carige Spa Gen & Imper ITA �0.10 2.39Banco Desio Della Brianza ITA �0.02 1.76Banco Popolare ITA �0.06 2.86Banca Popolare Di Milano ITA �0.03 2.78Banca Monte Dei Paschi Siena ITA �0.12 2.96

Bank of Siauliai Ab LTU �0.06 2.97

ING Groep Nv NLD �0.01 3.14Van Lanschot Nv NLD �0.03 1.62

Mbank Sa POL 0.05 2.34Bank Handlowy W Warzawie Sa POL 0.01 2.05ING Bank Slaski Sa POL 0.04 1.90Bank BPH Sa POL �0.09 4.48Bank Millennium Sa POL 0.03 2.62Bank Plsk Kasa Opk Grp Pekao POL 0.00 2.26Bank Zachodni Wbk Sa POL 0.04 2.15Getin Holding Sa POL �0.02 3.16Powszechna Kasa Oszczednosci POL 0.00 2.02

Banco BPI Sa PRT �0.03 2.46Banco Comercial Portugues Sa PRT �0.09 2.76

Svenska Handelsbanken SWE 0.02 1.86Skandinaviska Enskilda Bank SWE 0.01 2.55Nordea Bank Ab SWE 0.02 2.05Swedbank Ab SWE 0.01 2.53

The one-step-ahead forecast is OYtC1 D AYt . The forecast error is the differenceyi;tC1 � Oyi;tC1 D �ij;0wj;tC1, and its variance is therefore

var.yi;tC1 � Oyi;tC1/ DNX

j;kD1

�ij;0�ik;0 var.wj;tC1; wk;tC1/ DNX

j D1

�2ij;0: (4.2)





10 S. Tungsong et al

Each term �2ij;0 in the sum is interpreted as the contribution to the one-step forecast

error variance of variable i due to shocks in variable j . Its normalized value, cij D�2

ij;0=PN

kD1 �2ik;0

, is called connectedness by Diebold andYilmaz (2009, 2012, 2014)and it is associated with the relative uncertainty spillover from variable j to variablei . In this paper, we will report on the “total connectedness”, which is

total connectedness D 1

N

NXi;j D1i 6Dj

cij (4.3)

and measures the average effect that the variables have on the one-step forecast errorvariance. It is a measure of spillover uncertainty across the entire system. Larger valuesof total connectedness correspond to unstable periods in which variables’uncertaintiesstrongly influence one other.

We refine the original Diebold and Yilmaz (2009, 2012, 2014) methodology byintroducing two technical improvements. The first improvement consists in employ-ing ridge regularized VAR (Hoerl and Kennard 1970; Tikhonov 1963), which is usedto make estimations less sensitive to the noise and uncertainty associated with having atime series of finite length. Ridge regression introduces a penalty on the square sum ofregression coefficients, thus favoring models with smaller coefficients. This improvesregression performances, especially for systems with a large number of variables,where the covariance matrix is nearly singular (see Gruber 1998). In practice, ridgeregression consists in adding a diagonal term in the expression for the regression coef-ficients: B D .XX 0 C �I/�1XY 0, with I the identity matrix and � a coefficient thatmakes the inversion less sensitive to uncertainty over small eigenvalues (Tikhonov1963). The parameter � must be chosen with respect to regression performances; itdepends on the length of the time series and on their statistical properties. In ourcase, we used � D 100, which we determined was a good compromise value for thisdata set, and a window length of 300 points.2 We verified that the results are slightlysensitive to variations of � in a wide range Œ100–1000�. The second technical improve-ment consists in using exponential smoothing to mitigate the effects associated withsensitivity to large variations in remote observations (Pozzi et al 2012). Exponentialsmoothing computes weighted averages over the observation window, with exponen-tially decreasing weights, exp.�s=�/, assigned to more remote observations (here, s

counts the number of points from the present). In this paper, we use rolling windowsof size 300 days with exponential weights of characteristic length � D 100. Choosinga characteristic length approximately one-third of the window’s length was suggestedas optimal by Pozzi et al (2012).

2 We multiplied returns by a factor 100 in our analysis. Therefore, the value � D 100 is reasonablecompared with the norm of the matrix XX 0, which is of order 104.






4.2 Transfer entropy and Granger causality

We investigate how uncertainty in one region affects uncertainty in another regionby quantifying lead–lag relationships among uncertainty spillovers. For this purpose,we compute the transfer entropy associated with the daily and weekly changes in thetotal connectedness of the three systems.

In this paper, we estimate the transfer entropy by using both linear and nonlinearapproaches. The transfer entropy TY !X quantifies the reduction of uncertainty on thevariable X that is provided by the knowledge of the past of the variable Y taking intoconsideration the information from the past of X . In terms of conditional entropies,it can be written as

TY !X D H.Xt j Xt -lag/ � H.Xt j Xt -lag; Yt -lag/; (4.4)

where Xt represents the present value of variable X and Xt -lag its lagged past. In thispaper, we report the results for one-day lag. The conditional entropies are defined asH.A j B/ D H.A; B/ � H.B/, with H.A; B/ the joint entropy of variables A andB and H.B/ the entropy of variable B .

For what concerns the computation of these entropies, the linear approach is thestandard procedure. It quantifies the additional reduction in the variance of a variableY provided by the past of variable X , and it is directly related to Granger causal-ity (Barnett et al 2009; Granger 1988). In this linear case, the entropy associatedwith a set of variables Z is proportional to the log determinant of the covarianceH.Z/ D 1

2log det.2e�˙.Z//, where ˙.Z/ is the covariance matrix of the variables

in Z. The result of using (4.4) is that TY !X is simply given by half the logarithm of theratio between the regression error of variable X regressed with respect to Xt -lag and theregression error of variable X regressed with respect to both Xt -lag and Yt -lag. The non-linear approach instead estimates entropies by first discretizing the signal into threestates, associated with a central band of values within ı standard deviations from themean and two external bands, respectively, with values smaller or larger than the cen-tral band. By calling p0

A, p�A and pC

A the relative frequencies of the observations in thethree bands, entropy is estimated as H.A/ D �p�

A log p�A � p0

A log p0A � pC

A log pCA .

The joint entropies are equivalently defined by the joint combination of values of thevariables in the three bands, and the transfer entropy is retrieved by applying (4.4).

The information flow can be measured by comparing transfer entropies in bothdirections. If TY !X > TX!Y , then one can say that the direction of the informationis prevalently from Y to X ; conversely, if TX!Y > TY !X , then the direction of theinformation is prevalently from X to Y . The net information flow between X and Y

can be quantified as TX!Y � TY !X .






We validated the statistical significance of transfer entropy by comparing our resultswith the null hypothesis generated by computing 10 000 values of the transfer entropy,which in turn was obtained by randomizing the order of the lagged variables. Thisprovides a nonparametric null hypothesis from which p-values can be computed. Wealso compared this nonparametric p-value estimate with the one from F -statistics inthe linear case and found comparable results.

5 RESULTS

5.1 Total connectedness

Using data from January 2005 to October 2015, we compute the total connectednessof the three banking systems – NA, the EU and ASEAN – over a rolling window of300 days for the ten-year period from March 2006 to November 2015. Figures 1, 2and 3 report the results for each of the three systems, comparing the original approachof Diebold andYilmaz (2009) (dashed red lines) with the improved approach proposedin this paper (solid blue lines). Let us first observe that the two approaches demonstratesimilar values and comparable behavior with regard to total connectedness. We cansee that the use of ridge regularized VAR eliminates some of the outlying spuriouspeaks observed with the original method. The effect is present in all samples acrossthe three regions and periods, but it is particularly evident in Figure 2 for the peaksafter January 2011 and January 2012. When dimensionality is high, as in the case ofthe EU banking system, ordinary least squares estimates tend to have high varianceas a result of overfitting. Ridge regression provides parameter estimates that havelow variance across rolling windows, which is a manifestation of the model’s abilityto better generalize across different samples. This is why we observe no suddenjumps in the total connectedness when we estimate our VAR coefficients using ridgeregression. More evident is the effect of exponential smoothing, which makes peakssharper and eliminates the plateau effect due to the persistence of the influence ofa peak during the whole length of the rolling window. This is especially evidentin Figure 1, where for the standard VAR method the peak in total connectednessobserved just after January 2009 persists, creating a plateau that drops abruptly after300 days in January 2010. Conversely, the exponential weighted ridge regularizedmethod reveals a clear peak, reaching its maximum around January 2009, followedby a sharp decrease. We observe that the plateau effects in the standard VAR-equal-weights method sometimes completely hide peaks that are instead detected with theexponentially weighted ridge regularized method. This is the case for the late-2010NA spillover peak, which is visible in Figure 3 only for the exponentially weightedridge regularized method.






FIGURE 1 ASEAN banking system: comparison between total connectedness computedwith classical VAR approach (dashed red line) and the proposed approach (solid blue line),with ridge penalization and exponential smoothing.

Jan-06 Jan-08 Jan-10 Jan-12 Jan-14 Jan-160.2

0.3

0.4

0.5

0.6

0.7

Equal weights, standard VARExponential weights, ridge regularized VAR

Computations are over a 300-day rolling window.

Note that in Diebold and Yilmaz (2009), where total connectedness in both equityindex returns and equity index return volatilities was measured, the authors found thatthe return spillovers demonstrated “a gently increasing trend but no bursts, whereasvolatility spillovers display[ed] no trend but clear bursts”. Our results in Figures 1, 2and 3 indicate that applying exponential weights to the returns allows us to observeboth trends and bursts in the return uncertainty spillovers.

A comparison between ASEAN, EU and NA total connectedness from the ridgeregularized VAR models is presented in Figure 4, where major events are labeledon the graph when they occurred. The general shapes of the total connectednessof the three banking systems appear to be similar. Over the approximately ten-year period from March 2006 to November 2015, the values of NA’s total con-nectedness are generally higher than those of the EU and ASEAN banking sys-tems, except in the following periods: 2006 to mid-2007, early 2011, early 2013and mid-2014.






FIGURE 2 EU banking system: comparison between total connectedness computed withclassical VAR approach (dashed red line) and the proposed approach (solid blue line), withridge penalization and exponential smoothing.

0.4

0.5

0.6

0.7

0.8


Jan-06 Jan-08 Jan-10 Jan-12 Jan-14 Jan-16


The fact that NA, EU and ASEAN banking systems have different levels of inter-connectivity reflects the dissimilarities in the natures of the three banking systems. Ourdata set includes large banks operating at the national level (GICS code 40101010)that survived in the period from January 2005 to October 2015. Based on the GICScode and survival criteria, our NA system covers two countries (ten banks), the EUcovers seventeen countries (sixty-six banks) andASEAN covers five countries (thirty-nine banks). The two countries in the NA system (the United States and Canada) havebanking regulations that are more similar than those of the seventeen countries inthe EU or those of the five countries in ASEAN. In addition, the equities of the tenbanks in NA trade on the same stock exchange – the NYSE – while those of theEU and ASEAN banks trade on different national stock exchanges. Finally, as bankstend to form business relationships with other banks that are in close proximity, both






FIGURE 3 NA banking system: comparison between total connectedness computed withclassical VAR approach (dashed red line) and the proposed approach (solid blue line), withridge penalization and exponential smoothing.

Jan-06 Jan-08 Jan-10 Jan-12 Jan-14 Jan-160.2

0.3

0.4

0.5

0.6

0.7



geographically and from a regulatory perspective, the number of interbank businessactivities in NA is likely to be higher than in the EU and ASEAN. These three factorscontribute to stronger links and a higher possibility of spillovers among NA banksthan among EU or ASEAN banks. For the above reasons, total connectedness in theNA system is generally higher than in the EU and ASEAN systems.

The number of banks in a system does not seem to be a factor that influencesthe level of total connectedness, as there is no relationship between the number ofbanks and total connectedness in a system. Note that the total connectedness metricis computed on a per-bank basis; it is the average of all pairwise connectedness in asystem.

From visual inspection of Figure 4, we note that variations in total connectednessof the NA banking system seem to lead those of the EU and ASEAN systems, whilethe total connectedness of the EU system seems to lead that of the ASEAN system.This prompts us to perform causality tests on the total connectedness time series of






FIGURE 4 Total connectedness in the three banking systems (as in Figures 1–3; solidlines).

Jan-

06

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

B

CD E

FG

H I

J KL

N

O

P Q

R

EU sovereign debt crisisGlobal Financial CrisisSubprime

crisis

Jan-

11

Jan-

12

Jan-

13

Jan-

14

Jan-

15

Jan-

07

Jan-

08

Jan-

09

Jan-

10

Jan-

16

ASEAN spilloverEU spilloverNA spillover

A

M

Major events associated with peaks are indicated by letters in the figure:A, subprime mortgage crisis;B, securitizationmarket closedown; C, global stock market sharp fall; D, US near-record deficit US$410 billion; E, nationalization ofNorthern Rock;F, Fannie Mae/Freddie Mac rescue;G, Lehman Brothers filed for bankruptcy;H, rescue of RBS, Lloydsand HBOS; I, IMF approved US$2.1 billion loan for Iceland; J, US government gave Bank of America US$20 billionaid; K, RBS reported £2.1 billion loss; L, 12.5% economic contraction in Japan; M, Greek €120 billion bailout; N, UScredit downgrade from AC to A; O, London Interbank Offered Rate (Libor) scandal; P, City of Detroit bankruptcy; Q,Ukranian/Syrian/Egypt unrest; R, Ebola epidemic. Computations are over a 300-day rolling window.

the three banking systems in order to investigate how systemic uncertainty in eachregion influences the others as well as the lead–lag relationships among them.

5.2 Causality tests on regional total connectedness

In order to quantify the lead–lag relationships among the NA’s, EU’s and ASEAN’stotal connectedness, we compute transfer entropy and information flow between thedaily changes of total connectedness in the three regions for a one-day lag. Resultsare reported in Table 4. Transfer entropies are estimated using both the linear andthe nonlinear approaches discussed in Section 4.2. We recall that the linear measureis equivalent to Granger causality, where a significant transfer entropy correspondsto a validated Granger causality relation. The nonlinear measures are computed for






TABLE 4 Quantification of transfer entropy between regional total connectedness(March 28, 2006–November 2, 2015; full sample): from daily changes in the total con-nectivity using a one-day lag.

Netinformation

Method TENA!EU TEEU!NA flow

Linear 0.004722�� 0.001354� 0.003369Nonlinear threshold � 0.005251�� 0.006711�� 0.001460Nonlinear threshold 2� 0.003980�� 0.002012� 0.001968Nonlinear threshold 3� 0.004939�� 0.000561 0.004378

Netinformation

Method TENA!AS TEAS!NA flow

Linear 0.017336�� 0.008931�� 0.008405Nonlinear threshold � 0.008789�� 0.005837�� 0.002953Nonlinear threshold 2� 0.005348�� 0.002305� 0.003042Nonlinear threshold 3� 0.003150�� 0.002803�� 0.000348

Netinformation

Method TEEU!AS TEAS!EU flow

Linear 0.005659�� 0.003633�� 0.002026Nonlinear threshold � 0.005553�� 0.001262 0.004291Nonlinear threshold 2� 0.005960�� 0.000228 0.005732Nonlinear threshold 3� 0.004238�� 0.002118�� 0.002120

�p-value < 0.05. ��p-value < 0.01. ��p-value < 0.001.

fluctuation bands at ı D 1; 2; 3 standard deviations (see Section 4.2). Observe thatthere is a significant information transfer between NA and EU, NA and ASEAN,and EU and ASEAN that, for the linear case, implies NA Granger causes EU, NAGranger causes ASEAN, and EU Granger causes ASEAN. We observe that the non-linear estimation gives consistent results with the linear estimate for all values ofı, demonstrating the robustness of the result. We also observe that there are signifi-cant causal relations in the opposite direction. Given the extended time-lags betweenthe three regions, it is fair to question whether a one-day time lag and a one-daytime horizon will affect markets asymmetrically depending on their relative openinghours. We therefore test the flow of information across regions for a time horizon andlag of five days instead of one day. The results for the transfer entropies and infor-mation flow, performed for the entire period on nonoverlapping five-day returns, arereported in Table 5. We observe that the results are consistent with those for a one-day






TABLE 5 Quantification of transfer entropy between regional total connectedness (March28, 2006–November 2, 2015; full sample): from weekly changes (five days) in the totalconnectivity using a five-day lag.

Netinformation

Method TENA!EU TEEU!NA flow

Linear 0.008003� 0.001255 0.006747Nonlinear threshold � 0.009204 0.009474 �0.000271Nonlinear threshold 2� 0.017228�� 0.003196 0.014032Nonlinear threshold 3� 0.024087�� 0.002335� 0.021752

Netinformation

Method TENA!AS TEAS!NA flow

Linear 0.017200�� 0.003703 0.013497Nonlinear threshold � 0.010598� 0.004354 0.006244Nonlinear threshold 2� 0.006509 0.006475 0.000034Nonlinear threshold 3� 0.002107 0.006805�� 0.004698

Netinformation

Method TEEU!AS TEAS!EU flow

Linear 0.022020�� 0.000619 0.021401Nonlinear threshold � 0.021641�� 0.002374 0.019267Nonlinear threshold 2� 0.022964�� 0.002900 0.020063Nonlinear threshold 3� 0.007488�� 0.000405 0.007083

�p-value < 0.05. ��p-value < 0.01. ��p-value < 0.001.

time horizon and lag reported in Table 4, the main difference being the lower statis-tical significance. This is expected because the time series for the five-day changesare five times shorter than those for daily changes.

6 CONCLUSION

We investigate regional and inter-regional uncertainty spillovers in the NA, EU andASEAN banking systems during a period characterized by great regional and globalfinancial stress (2005–15). Uncertainty and financial instability is quantified by meansof total network connectedness, which we measure by improving on the methodof Diebold and Yilmaz (2009). We demonstrate that exponential smoothing andridge regression provide better-defined peaks in the temporal analysis and avoid the






occurrence of some spurious peaks. We observe that the NA system appears to beconsistently more interconnected than that of the EU, which in turn is more intercon-nected than the ASEAN network. Similarly to the previous analysis of Diebold andYilmaz of other systems, our empirical analysis of the NA, EU and ASEAN bankingnetworks shows that increased connectivity corresponds to periods of higher distressin the system. We observe that all large peaks of total network connectedness areassociated with identifiable major exogenous events. Despite some of these eventsbeing related to specific regions, the effects are seen across all three banking systems,which reveal similar patterns of peaks and troughs in the variations of their total net-work connectedness. However, such variations are not perfectly synchronous acrossthe regions, and causality patterns are discovered using transfer entropy. Our analysisreveals that the NA banking system is the most influential, having the largest effecton the other systems. However, feedback effects are measured with significant causalrelations in the opposite direction as well. The results are demonstrated to be robustwith respect to changes in the method used to compute the transfer entropy, changesin the values of parameters, and with respect to the use of daily or weekly returns inthe analysis.

To summarize, the contribution of this paper is threefold. First, we improve thetechnical aspect of VAR estimation, allowing for better identification of events con-centrated at specific times, which leads to a more accurate and insightful interpretationof the results. Second, we focus on connectedness in the banking sector, while pre-vious studies based on the Diebold and Yilmaz (2009) methodology have analyzednetworks of financial institutions. In particular, we analyze the NA, EU and ASEANbanking systems individually and show that, despite the regions’being geographicallydistant, they are affected to varying degrees by major financial crisis events originat-ing in dominant regions such as the NA and EU banking systems. Third, we perform acausality analysis on the regional connectedness time series generated using DieboldandYilmaz’s method. Our analysis suggests that a regional disaggregated investigationhas the advantage of introducing a predictive component to this methodology. Whilethe network total connectedness measure identifies increases in regional uncertaintyassociated with major events that shake the markets, the causality relation betweentotal connectedness in different regions – introduced in this paper – provides a quan-titative characterization of the flow of uncertainty from region to region, which couldbe interpreted as the result of contagion. To the best of our knowledge, this causalityanalysis is the first of its kind.

In future, we will compare this approach with other information theoretic measureswith the aim of finding a framework that is capable of qualifying financial uncer-tainty and its causal effects at all levels of aggregation, from a local single-variableperspective to a global world-market view.






DECLARATION OF INTEREST

The authors report no conflicts of interest. The authors alone are responsible for thecontent and writing of the paper.

ACKNOWLEDGEMENTS

T.A. and F.C. acknowledge the support of the UK Economic and Social ResearchCouncil (ESRC) in funding the Systemic Risk Centre (ES/K002309/1).

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Research Paper

The quest for living beta: investigating theimplications of shareholder networks

Matthew Oldham

Computational Social Science Program, Department of Computational and Data Sciences,George Mason University, 4400 University Drive, Fairfax, VA 22030, USA;email: [email protected]

(Received September 11, 2017; revised March 27, 2018; accepted April 17, 2018)

ABSTRACT

Network science is being increasingly utilized to assist in the search for causes ofirregular behavior in financial markets. The search gained greater impetus after tra-ditional finance theories were unable to predict the extent of the most recent globalfinancial crisis. The increased abilities of researchers to access and manipulate datahas also opened new avenues of investigation, including the discovery of key networksand the agents that interact within them. In this paper, an analysis of the temporal net-works formed between US institutional investors and Standard & Poor’s 500 stocksbetween 2007 and 2010 is presented, with the results identifying key relationshipsbetween the density of these networks and the movement of the market. The analysisalso identified the changing behavior of investors, as their risk aversion varied aheadof the market’s price movements. To a lesser degree, relationships between the returnof individual stocks and their investor networks are reported.

Keywords: social network analysis; complex systems; stock markets; finance; adaptive cycles;dynamic network analysis.

Print ISSN 2055-7795 j Online ISSN 2055-7809© 2018 Infopro Digital Risk (IP) Limited







26 M. Oldham

1 INTRODUCTION

The behavior of financial markets has frustrated investors and academics, and contin-ues to do so, resulting in the pursuit of new explanations. The pursuit of alternativeexplanations for the behavior of financial markets, and economics in general, hasbenefitted from interdisciplinary insights resulting from the development and evolu-tion of new disciplines such as econophysics. Within many of these new approaches,network science has become a key component, as it allows for the representation ofpatterns of connections and/or interactions within a given system (Newman 2010). Avital component of network science that is highly relevant to the varying behavior offinance markets is that the behavior of a system can vary greatly depending on thenetwork structure (the topology) of a system: a point investigated in this paper.

One approach that has benefitted greatly from the rise of network science is consid-ering financial markets as complex systems. The relevance of the complex systemsapproach is articulated by Sornette (2014), who concludes that, after twenty yearsof research, the key concepts required to understand stock market returns are imita-tion, herding, self-organized cooperativity and positive feedback, all of which can beassessed with the aid of network science. Another promising approach, which alsoutilizes network science, is considering the financial market as an ecosystem (see,for example, May et al 2008; Farmer 2002). One theory originating from ecologythat may be relevant to financial markets, and which is explored in this paper, is theadaptive cycle described by Holling (2001), the appeal being its ability to explainthe robustness and resilience of a system as a function of the resources and theirconnectedness within the system in question.

The improved ability of researchers to access, and then to analyze, extensivedata sets is another factor in the growth of interdisciplinary approaches. To achieve theresearch goals of this paper, an extensive data set collated from the Thomson Reuters13F database is utilized, along with other financial metrics, to undertake a temporalanalysis of the networks formed between US institutional investors and the stocks inthe Standard & Poor’s 500 (S&P 500) index. The analysis makes use of both pro-jected and bipartite networks to uncover numerous novel insights into relationshipswithin the market in general as well as between stocks and their shareholders. Onesuch insight explains how changes in a stock’s network connection are related to itsreturns, that is, its “living beta”.

The remainder of this paper is set out in the following manner. Section 2 provides abrief literature review, justifying the approach taken in this paper. Section 3 details theapproaches taken and the data used to undertake the analysis, while Section 4 providesthe results of the analysis. Both Sections 3 and 4 are subdivided into separate sectionscovering the bipartite as well as the investor-by-investor and stock-by-stock networks.A conclusion and summary are then provided in Section 5.





The quest for living beta 27

2 BACKGROUND

Financial markets are characterized by periods where price movements and tradingvolumes become much more volatile than expected. One of the most recent occur-rences of such an event saw the Dow Jones Industrial Average fall 21% in the first ninedays of October 2008, and the world subsequently plunged into the global financialcrisis (GFC). In response to the failure of traditional economic and finance theories toexplain such an event, Schweitzer et al (2009) argue that the analysis of economic net-works has become essential. Their rationale is that existing theories, and the policiesassociated with them, are inadequate for understanding and analyzing the interdepen-dencies that have always existed – and are continually growing – across global trade,supply chains and investment networks.

Since the existence of a network between investors was formalized by Shiller andPound (1989), an extensive body of work has been developed to understand the rami-fications of the opinions and actions of investors flowing across these networks. Thisincludes the ability of networks to explain investor trading decisions and portfolio per-formance (Ozsoylev and Walden 2011). The importance of networks within financialmarkets is further emphasized with the excess volatility seen in stock markets beingpartially explained by investors herding as they mimic investors in their network, aphenomenon that was first modeled by Cont and Bouchaud (2000). The essence of thisherding is captured in the research of Ben-David et al (2012), who find that “hedgefunds exited the US stock market en masse as the financial crisis evolved, primarilyin response to the tightening of funding by investors and lenders”.

Existing works relating to the network structures associated with financial marketshave tended to focus on forming networks based on the correlation

(1) between individual stocks (see Preis et al 2012; Bonanno et al 2004; Boginskiet al 2006; Kenett et al 2010) and

(2) between investors by implying an investor network (Ozsoylev et al 2014) orfinding an actual network (Shiller and Pound 1989; Hong et al 2005).

A characteristic of many of these studies is that the analysis therein has been performedon static networks, whereas the analysis of temporal networks (dynamic networkanalysis) will ultimately provide a greater insight into the behavior of the market.However, in an approach relevant to this paper, utilizing the correlation-between-stocks approach, Boginski et al (2006) assessed the temporal dynamics of the USstock market network between 1998 and 2002 and found that the market networkincreased in density over time, a fact Boginski et al (2006) attributed to increasedglobalization. The size of the maximum clique was also reported to have increasedover time, suggesting the stocks had an increasing tendency to move together.





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The approach taken in this paper extends the use of a bipartite network, that is, anetwork between stocks and investors, as seen in Caldarelli et al (2004). A benefitof this approach is that the relationship between a manager and the stocks in whichthey have invested is directly supported by the data, and as such does not need to beimplied through an alternate approach. The point of difference with this paper is thatthe network will be assessed over sixteen quarters, as opposed to a static network. Akey finding of Caldarelli et al (2004) was that their resulting market graph, in termsof the degree distribution with regards to the number of owners for companies, wascharacterized by a scale-free topology. This result provides evidence that financialmarkets operate as complex adaptive systems (CASs), meaning that traditional finan-cial theories based on equilibrium conditions are likely to be insufficient in explainingthe dynamics of financial markets.

To emphasize the point made by Schweitzer et al (2009) – namely, that traditionalmethods are inadequate in understanding the interdependencies seen in networks– the aforementioned findings regarding the degree distribution of the network areinconsistent with the capital asset pricing model (CAPM) proposed by Sharpe (1964).The CAPM predicts that investors will only be compensated for the time value ofmoney and the level of nondiversifiable risk (or systematic risk), with the rate atwhich investors should be compensated being captured in the beta coefficient. Thereremains, however, much debate around the accuracy and relevance of the CAPM(Fama and French 2004). Indeed, evidence against the CAPM utilizing a networkapproach is provided by Caldarelli et al (2004), who suggest that if the CAPM held,then stocks would record a constant in-degree. This outcome was not observed, thusproviding support to the pursuit of alternative approaches.

3 METHOD

3.1 Approach

The research undertaken in this paper explores the temporal aspects of the bipartite net-works (as well as the subsequent stock-by-stock and investor-by-investor networks),formed for each quarter between 2007 and 2010, of the stocks in the S&P 500 and theUS institutional investment managers that held them (note that the terms “investmentmanager” and “manager” are interchangeable with “investor” and have been usedas such in this paper). This period was selected as it covers the GFC and thereforeprovides a rich source of dynamics as investors reacted to the falling and then recov-ering market. The main contribution of this approach is an attempt to understand thenetwork dynamics across multiple periods rather than interpreting a static position ofthe network.






3.2 Data

The data used for our research was collected from two sources. The shareholdingdata was collected from the Thomson Reuters 13F database, which records the stockownership of all US institutional investment managers, and was accessed via theWharton Research Data Services website. A US institutional investment investor isdefined as an investment adviser, bank, insurance company, broker–dealer, pensionfund or corporation who utilizes the US mail service in the course of operating theirbusiness, and who exercises investment discretion over at least US$100 million inassets. Therefore, while not capturing all investors, the data does encompass approx-imately 65% of investors.1 The database provided the option of forming the networkbased on either the ID number or the name of the manager. The manager ID numberwas chosen because there was evidence of trivial differences in the manager namefrom period to period, causing the formation of unnecessary nodes across the sampleperiod.

It must be noted that the accuracy of the Thomson Reuters 13F database is notbeyond question. Anderson and Brockman (2016) raise several objections regardingits accuracy, before speculating that the most significant errors most likely lie in themisreporting of the value of period ending holdings. However, according to Ben-David et al (2015), the error rate grew materially after 2013, which is beyond thescope of the data utilized in this research.

The greater issue pertaining to this paper is the accuracy of the number of sharesheld. Regarding this, Lewellen (2011) quantifies the issue of more stocks beingrecorded than shares on issue at 1%, with the quantity of reports shares in excessof the total shares being 5% or less in 50% of all cases, thus providing a level of com-fort that the data errors are not material.As mentioned below, significant preprocessingof the data was undertaken to avoid errors.

In terms of the filing requirements for investment managers, Rule 13f-1(a)(1) of theSecurities Exchange Act of 1934 (15 US Code 78m(f)) requires managers to submitfour Form 13F filings once they meet the US$100 million filing threshold on thelast trading day of any month during any calendar year. The rule also requires thatmanagers file four Form 13F filings, even if after meeting the US$100 million filingthreshold they fall below the threshold in subsequent periods. The requirements ofthis rule will limit the influence of any survivorship bias in the sample set, becausemanagers will still be required to report regardless of whether their funds undermanagement took a temporary hit during the GFC.

The shareholding data did require considerable preprocessing, including the iden-tification and removal of duplicated records. In addition, stocks that did not have

1 Based on www.heritage.org/taxes/report/most-stocks-are-held-private-investors.





30 M. Oldham

records for each of the sixteen quarters, because of a takeover or bankruptcy, forinstance, were removed. In summary, the data comprised over 3.6 million records,with approximately 3600-plus distinct investors identified for 477 stocks, noting thatthe number of investors in each period varied materially. It should be assumed that ifan investor ceased reporting, it was due to them merging, falling below the US$100million threshold for an extended period, or no longer holding the S&P 500 stocks inthe sample.

The shareholding data provides the benefit of being able to generate two networkframeworks. The first is a binary network, where a link is formed and has a weightof 1 if an investment manager holds a given stock. The second is where the weight ofthe link is determined by the manager’s proportional holding in a given stock, as per(3.1). It should be noted that the denominator in (3.1) is the sum of the shares fromthe data rather than the total shares on offer for the stock. Future iterations of thisanalytical framework may look to utilize the alternate approach

wijt D SijtPJj Sijt

: (3.1)

Here, wijt refers to the weight of the link between the i th stock and the j th managerin quarter t ; Sijt is the number of shares held by the j th manager in the i th stock inquarter t ; and

PJj Sijt is the sum of all the managers’ (J ) holdings in the i th stock

in quarter t .The financial data (price and financial metrics) was collected from the Thomson

Reuters Datastream system. The pricing data was used to create a return series for eachof the stocks (see Section 4.3) and to calculate the betas, as per the CAPM, for each ofthe stocks in each of the forty-eight months in the sample. The betas were calculated asper Yahoo Finance’s definition, which requires regressing thirty-six monthly returnsfor the stock in question against the returns of the S&P 500 index for the comparableperiod. The financial metrics were collected to assist in identifying the characteristicsof the communities, as identified in Section 4.3. The metrics, and the rationale fortheir use, are as follows.

� The price-to-book (PB) ratio, which compares the market value of a stock withits book value, as provided in the stock’s financial statements. It is calculatedby dividing the current closing price of the stock by the latest quarter’s bookvalue per share. A lower PB indicates that a stock is possibly undervalued andis therefore a candidate for a value manager.

� The price-to-earnings (PE) ratio, which is a measure of a stock’s current shareprice relative to its per-share earnings. Generally, a high PE ratio is an indicationthat investors are anticipating higher growth in the future.






� Market capitalization (market cap), which is the total market value in dollarsof a company’s outstanding shares. It is calculated by multiplying a company’soutstanding shares by the current market price of one share. The metric is usedto determine a company’s size, and to categorize a stock as either a large or asmall cap stock.

� Dividend yield (DY), which is obtained by dividing the dollar value of a stock’sdividend per share by the stock’s share price. In general, a higher DY indicatesan ex-growth stock since the company does not need the funds for future growth.

3.3 Network analysis

The collected data set allowed for analysis to be undertaken at a bipartite level aswell as in single-node networks. Justification for undertaking analysis at the bipartitelevel may be found in Opsahl (2013), where the difficulties of projecting a two-nodenetwork onto a single-node network are discussed. The main concerns put forwardare

� that the creation of links for the projected network is not consistent with howthey would be formed in a single-node network, with the contention being thatthe links would neither be independent nor formed in a random manner, and

� that the projected network will have larger, more fully connected cliques.

The downside to the bipartite approach is that many traditional network analysismetrics and techniques, such as betweenness and closeness centrality, are not suitedto being used in this manner. To overcome this, the single-mode networks (eitherinvestor-by-investor or stock-by-stock networks) were formed utilizing matrix alge-bra. Traditional analytical tools were then applied to these single-mode networks usingthe line of investigation detailed in Section 3.3.2, generating the results presented inSections 4.2 and 4.3.

3.3.1 The bipartite approach

To undertake analysis at the bipartite level, the R bipartite package (R Core Team2017; Dormann et al 2008) was utilized. Using the network level function, the clustercoefficient, connectance and links per species were selected to investigate the charac-teristics of the network. Using the approach outlined in Opsahl (2013), the clusteringcoefficient is returned for the overall network as well as for the upper and lower lev-els. The coefficient for a given level is derived from the (weighted) average clustercoefficients of its members. Connectance is defined as the sum of links divided bythe number of companies and multiplied by the number of investors, effectively, the





32 M. Oldham

density of the network. Finally, links per species is the average number of links forthe companies and investors.

An important point relating to these variables is that they are not directly affectedby variation in the number of managers reporting in each period. The rationale forthis is that the metrics are measures of actual links compared with possible links foreach independent period; in other words, they describe how connected the current setof managers is to the current set of stocks.

3.3.2 The single-mode approach

The single-mode networks, ie, stock � stock (S � S) and investor � investor (I � I),were created using both the binary and weighted bipartite networks (as described inSection 3.2). The relevance of the S�S network is that it generates a network in whichthe weight of the link between any two stocks represents the number of managers thathave a common holding in the two stocks. The creation of the S�S network resulted ina fully connected graph because, among the 2500-plus managers in any given period,at least two managers had a common shareholding in all the stocks in the data set.Given the extreme density of the network, the use of traditional network metricswas somewhat hindered and visualization of the network was pointless. However, asdetailed in Section 4.3, the networks did provide some novel and useful insights.

In contrast with the S � S network, the I � I network forms a link between twomanagers if they have a common shareholding in a given stock. While high, the densityof this network did not reach the saturation levels seen with the S � S network. Thedensity of the network follows a distinct pattern that is in line with the findingsshown in Figure 1; that is, the peak in the network density occurs ahead of the marketdecline, as it descends into the GFC, before recovering ahead of the market. Thedensity behaved in a similar manner during the market correction in mid-2010.

4 FINDINGS

4.1 Bipartite network

Illustrated in Figure 1(a), which plots the S&P 500 index and the cluster coefficient ofthe bipartite network, is the fact that the connectedness of the market system declinesahead of the market before recovering as the market recovers. Prior to further dis-cussion, it should be acknowledged that the process of higher (lower) connectednessbetween investors may not increase (decrease) the market capitalization of the market.If it is the case that the existing investors are simply diversifying their holdings, thenet effect should see no change in the market index. This result would be significantin and of itself since, in moving toward similar portfolios of stocks, it signifies that






investors are also moving toward homogeneous strategies, that is, they are forminglarger herds.

For connectedness to increase along with the market capitalization, two conditionsmust be met. First, new funds must be invested in the market. This might occurthrough a change in asset allocation by investors, the reinvestment of dividends and/or an increase in leverage. Second, and more importantly, as investors invest theiradditional funds they diversify their holdings rather than maintaining conviction intheir existing portfolios. Once again, this might suggest investors are moving towardmore homogeneous strategies.

The relationship between the S&P 500 index and the linkage density and con-nectance coefficients of the bipartite network show similar characteristics. The mostfeasible explanation for this result is that investors initially reduce their holdings bymoving to cash and/or focusing on quality stocks as the market falls. The clustercoefficient reaches its minimum point in the fourth quarter of 2008. It then recov-ers as investors return to the market with increased confidence following the variouspolicy initiatives from governments and central banks. Increased clustering occurs asinvestors spread their bets across the market, a point that is explored further in thefollowing section. Exiting and reentering the market is a classic example of herdingbehavior and has been captured clearly by movement in the network’s clustering coef-ficient: a trend that has gone unreported until this paper. Consistent with the market’sbehavior throughout 2008 and 2009, at the start of 2010 there was a correction, whichled to concurrent declines in the bipartite clustering coefficient and network densityin the I � I network.

The question of what is driving this process is partially explained in Figure 1(b) andFigure 1(c). The clustering coefficients for the stock and investor networks behavein very different manners. The stock network more closely resembles the overallnetwork, showing a decline in clustering that occurs ahead of the collapse in the S&Pindex. As the clustering coefficient is equivalent to the actual number of links dividedby the possible links, this finding tells us that investors dropped their holdings incertain stocks before picking them up again when the market recovered. This processis further investigated in Section 4.3. In contrast with the overall market network, thisclustering of investors grows almost unabated throughout the sample period. Thisis suggestive of the fact that, over the period in question, the trend for the investorpopulation was to “buy the market”, that is, for managers to diversify across a greaternumber of stocks, or at least buy a common set of safe stocks, rather than attempt toidentify stocks other than those that appeal to the “herd”.

A possible explanation for how and why the market responded in this fashioncomes from ecology and the model of the adaptive cycle (Holling 2001). The appealof this theory is that it focuses on the processes of destruction and reorganization,thus providing a more complete view of the system dynamics as it links system





34 M. Oldham

FIGURE 1 The S&P 500 and clustering coefficients of the bipartite network from 2007 to2010.

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(a) The movement of the S&P 500 and the cluster coefficient of the bipartite network between 2007 and 2010.(b) The weighted cluster coefficient for the stock level network. (c) The weighted cluster coefficient for the investorlevel network.

organization, resilience and dynamics. A key implication of the model is that, asa system grows, the components within the system will become more connected,increasing the output of the system. For the stock market, this means investors andcompanies become more connected, with a resulting increase in market capitalization.However, as the system becomes more connected, it also becomes less resilient to anyexternal shocks and may fail following a shock. A collapse will see the connectednessand output of the system decline, before it eventually reconnects in the recovery phase.This phenomenon is illustrated in Figure 1, where the clustering declines in unisonwith the value of the index.

The takeaway from this analysis of the investment market network is that investorsand market observers should expect to see the density of the bipartite network increaseto a certain point, at which time it will become susceptible to a shock. If the shockis sufficient, then investors will dissipate as they reduce their holdings before recon-necting during the recovery. This approach leaves open the question of how largethe shock needs to be to fracture the network. The level of connectedness could help






explain why the markets capitulated with the collapse of Lehman Brothers (and notBear Stearns, where the market was not as connected), recovered and then experiencedfurther corrections once the market became more connected.

The question of whether the findings shown in Figure 1 are pure coincidence –or whether the system did indeed behave in accordance with the adaptive cycle –is difficult to answer definitively. In support of the argument that financial marketsdo, in some way, function in accordance with the adaptive cycle, a simple review ofmarkets between 2007 and 2012 shows the occurrence of several periods of increasedvolatility associated with market corrections since the GFC. These corrections weregenerally the result of concerns either relating to a change in central bank policyor around sovereign debt issues in Europe. As previously noted, the first post-crashcorrection (mid-2010) saw concurrent movements in the relevant network structure;thus, subsequent corrections might be expected to lead to similar behavior.

The bipartite package produces cumulative degree distribution charts for each net-work level, stock and investor, and fits a power-law function to them. The relevanceof testing for the presence of a power law comes from Boginski et al (2006) andCaldarelli et al (2004), who both found that the degree distribution in their networksfollowed a power law, with the ramifications being that financial markets operateas CASs. In general, the distribution was heavily skewed across the sample periodbut did not meet the definition of a power law. A more detailed analysis of thedegree distributions and how they change over time is contained in the followingsections.

4.2 Investor analysis

An alternate interpretation of the degree centrality distribution of the I � I network isprovided in Figure 2. Via box plots for each of the sixteen periods in the sample, thefigure presents the degree centrality distribution calculated for the I � I network andshows that, consistent with the previously mentioned findings, the distribution is heav-ily skewed. However, the interpretation is slightly different, in that most managersare linked to many other managers, while a smaller number of managers are linkedto just a few managers. The key observation is that the range of the degree distribu-tion metric contracts after the bottoming of the market, suggesting that institutionalmanagers were more likely to share common holdings. This outcome is consistentwith managers’ tendency to herd, as they rotated into a common set of less riskystocks.

An alternate interpretation of degree centrality is illustrated in Figure 3, where thecentrality measure is calculated in accordance with (4.1). The variable measures thenumber of stocks held by each manager in each period, in other words, the in-degree





36 M. Oldham

FIGURE 2 Box plots illustrating variation in the degree centrality from 2007 to 2010, ascalculated for the I � I network.

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from the stock level in the bipartite network:

djt DPI

i Sijt

It

: (4.1)

Here, djt refers to the in-degree for the j th manager in period t ; Sijt is 1 or 0, depend-ing on whether the j th manager holds the i th stock in period t ; and

PIi Sijt denotes

the sum of stocks the j th manager holds in period t . The denominator normalizes thedegree count such that, if a manager held all 477 stocks in the sample for period t ,djt would be 1.

While the metric does not display the same variability as seen in Figure 2, it showsa skewed distribution, indicating that most managers hold fewer stocks. Nonethe-less, some managers hold many (noting that outliers are not shown in the chart). Touncover the dynamics of how managers varied their holdings throughout the period,Figure 3(b), which shows the standard deviation of the degree distribution, and Fig-ure 3(c), which shows the mean degree distribution for the managers, are provided.From Figure 3(b), it appears that most managers did not change the number of stocksthey held materially across the sample. In terms of the average number of stocks held,Figure 3(c) highlights the presence of index funds that would have held all the stocksin the index, with most managers holding 10% of the possible stocks.

Our initial impression from Figure 3(a) is that the change in the degree centralitymeasure across the sixteen periods is comparable with that seen in Figure 2. Furtherinvestigation is provided in Figure 4(a), which shows the change in the mean numberof stocks held by all managers. Consistent with the other findings in this paper, theaverage declines in periods of increased market volatility before recovering as themarket improves. With the range of the average holding per manager being five, somemay not assess this as being material, given the disruption in the markets. However, itshould be acknowledged that the stocks in the sample are from the S&P 500 and gen-erally reflect more stable stocks, given the larger market capitalization. Provided the






FIGURE 3 The variation and distribution of investor degree centrality from the marketbipartite network.

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(a) The variation in the degree centrality for investors as calculated by (3.1) in the bipartite network. (b) and (c) Thestandard deviations and means, respectively, of the centrality measure across the sample.

metric indicates only whether or not managers still hold stocks, rather than reflectingthe weighting of each of the stocks, an analysis of weighted holdings may ultimatelyprovide greater insight.

Further, no attempt was made to divide the investor population. Ben-David et al(2012), among others, provide evidence that the reduction of exposure by hedge fundsin this period was in excess of what could be interpreted from Figure 4(a). The dynamicis partially captured in Figure 4(b), which provides the standard deviation in terms ofthe number of stocks held by each manager. Consistent with Figure 1(c), it increasesin a monotonic fashion. Our interpretation is that investors were diverging in theirapproaches, with one side either maintaining or decreasing its number of holdings,while the other increased theirs, as they adopted the strategy of “buying” the market.This dynamic may be an indication that underlying investors were moving towardinvestment managers who were tracking the market.

4.3 Stock analysis

This section investigates more specifically how the network(s) pertaining to the stocksvaried over time. In a similar manner to Figure 3, Figure 5 provides information about





38 M. Oldham

FIGURE 4 The dynamics of the number of stocks held per investor and the number ofinvestors per stock.

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how the in-degree (a link from a manager) for each stock contained in the bipartitenetwork varied. The metric was calculated via

dit DPJ

j Sijt

Jt

: (4.2)

Here, dit denotes in-degree for the i th stock in period t ; Sijt is 1 or 0, depending onwhether the i th asset is held by the j th manager in period t ; and

PJj Sijt is the sum

of managers holding the i th asset in period t . The denominator normalizes the degreecount such that, if a stock is held by all the managers in the sample in period t , dit

would be 1.From Figure 5(a), there appears to have been little movement in the number of

managers who held each stock, with Figure 5(c) showing the average in-degree cen-trality for all 477 stocks. It does appear that there are some heavily owned stocks(that is, stocks with a high degree of distribution) in the index. The lack of volatilityis supported by Figure 5(b), which shows that the standard deviation in the in-degreedistribution for the 477 stocks was not large. The lack of variation is consistent withwhat was seen in Section 4.2, and a clearer picture of what is occurring can be seenin Figure 4(c). The chart illustrates that stocks during the GFC, on average, had areduced number of investors, as investors reduced their holdings in favor of quality






FIGURE 5 The variation and distribution of stock degree centrality from the market bipartitenetwork.

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stocks or cash. This finding is ultimately the same as what was seen in Figure 4(a).However, there is a contrasting finding in terms of the standard deviations. Figure 4(d)shows that variation in the number of investors in a stock declined during the GFCbefore recovering, something not seen in Figure 4(b). Interestingly, toward the end ofthe sample period the standard deviation grows strongly, again suggesting that certainstocks were heavily favored.

Having established the S � S networks, an attempt was made to detect a com-munity structure in these networks. Ideally, the network would be split into clearcommunities, with each stock allocated to a particular community based on one ormore of the following characteristics: industry classification, growth profile, divi-dend yield, and/or volatility. Alas, the outcomes were not as clear cut. An issue thatproved problematic in the community detection was the density of each of the six-teen S � S networks. This meant that the igraph function (Csárdi and Nepusz 2006),cluster_fast_greedy, which attempts to define communities via directly opti-mizing a modularity score, found little variation with which to split the stock. Greatersuccess, in terms of defining a greater number of communities, was achieved by usingS � S networks that were weighted by the proportion of shares held by each manager.





40 M. Oldham

The result of the community detection was that four communities were foundin each of the sixteen periods. Community 2 was the dominant community, withapproximately 42% of stocks allocated to it. Community 4 was the smallest, withapproximately 2.5% of stocks allocated to it, with several periods over 3%. In termsof intercommunity movement, the results indicated that financial and industrial stocksjoined community 1, while consumer services and technology stocks moved to com-munity 3. The exact dynamics behind these changes are unclear, but, with a viewto understanding this classification better, a further interpretation of the communitystructure is attempted, with communities subdivided according to the financial vari-ables (see Section 3.2). The results suggest that community 4 is a large cap growthcommunity, with the justification being its higher average beta, market value, PE ratio,PB ratio as well as its lower dividend yield. Community 1 may be value oriented incomparison with community 3, given its lower PB ratio and higher dividend yield.Further work in this area may look to extend this analysis, as the identification ofclear set communities will aid portfolio diversification.

4.4 Stock pricing behavior

Having reported the dynamics regarding the degree distribution of both stocks andinvestors, this section seeks to uncover any relationship between the price movementsof the stocks and their position in the network. The first step in this process was togauge how a stock should have performed. This was achieved by calculating the betas,as per the CAPM, for each of the stocks during each of the forty-eight months in thesample. While the metric is not without issues, it does provide a meaningful measureof how a stock’s returns compare with the market, with a beta greater (less) than onesuggesting returns (positive or negative) greater (less) than that of the market but inthe same direction.

Figure 6(a) provides box plots for each of the forty-eight months and, in combi-nation with Figure 6(b) and Figure 6(c), provides useful insights. What is seen isthe range of beta contracts as the market heads into the GFC. However, it is duringthe recovery period that the analysis uncovers the most significant insights. The firstpoint of note is that the median beta remains constant from March 2009 onward, withthe range tapering from that point. The range of returns for the stocks (displayed inFigure 6(b)) also contracts over this period. A possible mechanism for this outcomeis given by Figure 6(c), where we see the cluster coefficient at the investor levelincrease in an almost monotonic manner. This suggests that managers on the wholewere broadening their portfolios, in effect “buying the market” as the market rallied,and thereby reducing the heterogeneity among holdings and returns.

To understand whether network metrics are capable of explaining stock returns,several exploratory steps were undertaken. The first was to ascertain whether a stock’s






FIGURE 6 Box plots illustrating the variation in stock betas (calculated via the CAPM)and variation in stock returns for the sample period.

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mean eigenvector centrality (from the S � S networks) across the sample period wasrelated to its mean returns (as depicted in Figure 7(a)). The standard deviations werealso compared (see Figure 7(b)). The rationale behind this approach is that a high meaneigenvector centrality measure may be interpreted as an indication that managers withco-holdings in the given stock also possess co-holdings in other well-held stocks. Thismeans the stock is being used to balance a portfolio and is likely to be held by “core”managers, that is, ones who track the market. Therefore, the stock should show lessvolatility, as it is being held with other commonly held stocks. Unfortunately, thereappears to be little in the way of a meaningful relationship between a stock’s returnand its eigenvector centrality.

The second approach was to see whether a stock’s mean degree centrality (as definedby (4.2)) across the sample period was related to its mean returns (Figure 7(c)). Therationale behind this approach is as follows: the interpretation of the degree centrality





42 M. Oldham

FIGURE 7 Scatter plots detailing the relationships between the average (and standarddeviations of) stock returns and their mean degree distribution and eigenvector centrality.

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(a) Mean eigenvector�mean return per stock. (b) Mean eigenvector�standard deviation per stock. (c) Mean degreedistribution � mean return per stock. (d) Mean degree distribution � standard deviation per stock.

in this framework is that a stock with a consistently high value tends to be held bymany managers, thereby identifying it as a core holding or market darling. Therefore,a stock with a high mean value should show less volatility, as managers will alwayshold the stock. Alternatively, a stock that fluctuates between a low and a high degreecould be classified as a fade stock, with dramatic increases suggesting that trade inthe stock has become “crowded” as the market herds.

There does appear to be a stronger relationship regarding stocks with higher meandegrees experiencing lower average returns. This warranted further investigation toestablish whether the lower average returns were the result of larger variations intheir returns or more consistent (although lower) returns. The results of this analysisare illustrated in Figure 7(d), where there is a negative relationship (albeit with anR-squared of only 5%) in terms of the price volatility of a stock, as given by thestandard deviation of its returns (indicated along the y-axis in parts (b) and (d) ofFigure 7), and the average number of investors. The justification is that stocks beingheld by more managers are likely to see less volatility, since no individual managerhas sufficient power to move the price of the stock.






FIGURE 8 The summarized results of the regression analysis of the individual stockreturns and their changes in degree centrality.

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(a) The range of changes in the quarterly degree distribution and share price for the stocks in the sample. (b) Thecoefficients from regressing quarterly changes in degree distribution against the quarterly change in price. (c) TheR-squared returns from those regressions.

To further investigate the findings shown in Figure 7, linear regressions comparinga stock’s quarterly change in price (ie, its returns) and degree distribution were runfor each of the 477 stocks, with the interpretation of the slope being the “living beta”for the stock. The results are summarized in Figure 8(b) and Figure 8(c), which showa generally positive relationship between the changes in a stock’s price and in itsdegree distribution. This is a relatively straightforward interpretation, that is, as moreinvestors hold a stock, the price goes up (and vice versa). Unfortunately, as can beseen in Figure 8(c), the “living betas” do not provide a sufficiently strong explanationof a stock’s price movement.

Figure 8(a) provides a summary of the variation in price compared with the variationin degree distribution. Consistent with the “living beta” coefficients being at least 1,there was far more variation in price than in the degree distribution. A couple ofinteresting questions stemming from this figure are the following.





44 M. Oldham

� Why is the variation so much larger for price?

� What is occurring outside the US institutional investor network?

Regardless, the analysis shown in this section can identify fade stocks as well asperiods in which investors act with a herding mentality.

5 SUMMARY AND CONCLUSION

There is little doubt that the traditional economic/finance equilibrium solutions devel-oped in the 1950s and 1960s have been unable to explain periods of excess volatilityin global financial markets. This has led to numerous other approaches gaining promi-nence, as they have provided greater insight. One such approach entails consideringstock markets as CASs (see, for example, Johnson et al 2003; Farmer et al 2012).Within this framework, the importance of networks has been identified and has ledto extensive research utilizing network science. This field of research has consideredand found numerous factors that can explain the behavior of financial markets, withthese factors being beyond the realm of standard equilibrium analysis.

This paper presents a novel result, which was to capture the parallel movement inthe density of the investment network and in the volatility and value of the S&P 500index. This evidence suggests that the market may indeed function in a similar fashionto an ecosystem, with the adaptive cycle theory (Holling 2001) providing a feasibleframework. Further work in this area should look to make better use of weightednetworks and integrate trading volumes, which would provide further insight intohow rapidly the network is changing. Another step might include forming networksbased on buyers and/or sellers rather than simply the holders in any given month.While the results in terms of finding the “living beta” of stocks were not sufficient tochallenge established finance theories, the analysis did provide useful insights as wellas avenues for future research. The first priority in extending this line of research willbe to extend the data set and, if possible, increase the granularity of the data, becauseas it stands the quarterly data may be missing some dynamics. Another extension willbe to try to understand why some stocks experience greater fluctuation in their degreedistribution and to be aware of the tipping points when a trade in a stock has becometoo crowded.

The exploratory steps undertaken in this paper have confirmed the utility of networkdata in uncovering the dynamics of the stock market. The future for this line of researchis bright, and the continued fostering of an interdisciplinary approach will maxi-mize the probability of researchers identifying and forming appropriate responses tomarkets that are potentially overvalued and susceptible to a major correction.







The author reports no conflicts of interest. The author alone is responsible for thecontent and writing of the paper.

ACKNOWLEDGEMENTS

The author would like to thank the anonymous reviewers for providing valuablecomments and suggestions that were used to improve the quality of the paper. Iwould also like to thank George Mason University for funding me throughout thePresidential Scholarship program as well as my advisor, Professor Rob Axtell, forproviding insight and guidance in the preparation of this paper.

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Research Paper

A stock-flow consistent macroeconomicmodel with heterogeneous agents:the master equation approach

Matheus R. Grasselli and Patrick X. Li

Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton,ON L8S 4L8, Canada; emails: [email protected], [email protected]

(Received April 25, 2017; revised November 2, 2017; accepted May 2, 2018)

ABSTRACT

We propose a mean-field approximation to a stock-flow consistent agent-based macro-economic model with heterogeneous firms and households. Depending on their invest-ment elasticity to past profits, firms can be either aggressive or conservative. Con-versely, households are divided into investor and noninvestor groups, depending onwhether or not they invest a portion of their wealth in the stock market. Both firmsand households dynamically change their type according to transition probabilitiesspecified exogenously. The mean-field approximation consists of homogenizing thebalance-sheet variables for agents (firms or households) of the same type and comput-ing the time evolution of the corresponding average as a combination of the determin-istic dynamic, derived from investment and consumption decisions before a changeof type, and the probabilistic change in type, with an appropriate rebalancing to takestock-flow consistency into account. The last step of the approximation consists inreplacing the underlying Markov chain with a continuous-time diffusive limit. We

Corresponding author: M. R. Grasselli Print ISSN 2055-7795 j Online ISSN 2055-7809© 2018 Infopro Digital Risk (IP) Limited







48 M. R. Grasselli and P. X. Li

present numerical experiments showing the accuracy of the approximation and thesensitivity of the model with respect to several discretionary parameters.

Keywords: agent-based models; heterogeneous agents; stock-flow consistency; mean-field approx-imation.

1 INTRODUCTION

The distinction between the actions of individual agents and aggregate behavior hasbeen a central theme in macroeconomics at least since the work of Keynes, who inKeynes (1936) stated that:

For although the amount of his own saving is unlikely to have any significant influenceon his own income, the reactions of the amount of his consumption on the incomes ofothers makes it impossible for all individuals simultaneously to save any given sums.Every such attempt to save more by reducing consumption will so affect incomesthat the attempt necessarily defeats itself. It is, of course, just as impossible for thecommunity as a whole to save less than the amount of current investment, sincethe attempt to do so will necessarily raise incomes to a level at which the sumswhich individuals choose to save add up to a figure exactly equal to the amount ofinvestment.

Given the inherent challenges in assessing the individual behavior of a large num-ber of agents, Keynesian economics tended to focus instead on direct modeling ofaggregate variables, such as total savings and output. The difficulty with this approachis that it downplays the role of individual decision making, particularly in the face ofuncertainty. At the opposite end of the spectrum, the predominant dynamic stochasticgeneral equilibrium (DSGE) models of contemporary macroeconomics advocate thatall aggregate relationships need to be derived from individual decision-making agents:this is known as microfoundations. The problem with this position, however, is that, asa consequence of the celebrated Sonnenschein–Mantel–Debreu (SMD) theorem (see,for example, Mantel 1974), the hypothesized properties of individual agents (namely,intertemporal utility maximization) are generally not enough to guarantee that theresulting aggregate behavior (namely, general equilibrium) is stable. To circumventthis fundamental difficulty, DSGE models typically assume that each relevant sec-tor of the economy consists of a single representative agent, thereby avoiding theaggregation problem associated with the SMD theorem. Naturally, this simplificationalso throws away any possibility of emerging behavior arising, as the aggregate andindividual levels are automatically assumed to be identical: a weakness that has beenwidely cited as the core reason for the poor performance of DSGE models during therecent crisis (see, for example, Kirman 2010).





A stock-flow consistent macroeconomic model with heterogeneous agents 49

In recent years, several papers have attempted to extend DSGE models to incor-porate heterogeneous agents, starting with the seminal contributions of Krusell andSmith (1998) for heterogeneous households and Khan and Thomas (2008) for hetero-geneous firms. A recent survey of such models can be found in Ragot (2018), where itis explained that heterogeneity is introduced through a series of idiosyncratic shocksexperienced by different agents. The shocks can represent, for example, different lev-els of employment income for households or different levels of capital productivityfor firms, and they are typically modeled by a finite-state Markov chain with constanttransition probabilities. Beyond these exogenous shocks, agents are still considered tobe identical with respect to their decision making. For example, in Krusell and Smith(1998) all households have the same utility function, whereas in Khan and Thomas(2008) all firms are profit optimizers. In other words, as well as suffering from tech-nical problems of their own, these models fail to address the lack of heterogeneity inbehavior that is common to all DSGE models as a consequence of the SMD results.1

An alternative to both aggregate-level Keynesian and representative-agent-basedDSGE models in macroeconomics are agent-based models (ABMs), where agentsare not constrained by utility-maximizing behavior and aggregation is not achievedthrough equilibrium. The literature on these models has burgeoned since the 2007–8crisis, and a recent assessment of the results, including a comparison with DSGEmodels, can be found, for example, in Fagiolo and Roventini (2016). A commonobjection to ABMs is that they typically rely almost exclusively on numerical simu-lations, making them both computationally intense and difficult to interpret. This isparticularly acute when an ABM, as is often the case, has many underlying parame-ters. In the absence of a faster way to simulate the model, parameter estimation, forexample, can become prohibitively slow. One way to address this problem is to intro-duce semi-analytic approximations by way of mean-field interactions, as advocatedin Gallegati and Kirman (2012), for example.

The general mathematical framework for the application of mean-field approxi-mations of this kind to economics can be found in Aoki (2002). An application toa specific model explaining business cycle fluctuations is presented in Di Guilmiet al (2010) and Delli Gatti et al (2012). The key feature of the approach consistsof dividing the relevant sectors (say, firms or households) into types according tosome classification. Agents in the same type are then deemed to behave in a similarway (say, with respect to investment or savings), so that one can keep track of aver-ages (or other statistics) of the variables of interest, instead of their values for each

1 For example, strictly speaking, as observed in Ragot (2018), they are not DSGE models, sinceeach realization of the sequence of shocks gives rise to a new agent. Several approximations are thenused to “solve” the models in a way that resembles their motivating DSGE core, such as truncatingthe history of shocks to a fixed number of past time steps.






individual agent. Crucially, the agents are also assumed to make decisions based onthese averages, in what is called a mean-field interaction, rather than by direct inter-action with other agents. Finally, agents are allowed to change type in a probabilisticmanner, so that the time evolution of the distribution of agents is governed by theso-called master equation. This achieves considerable simplification by replacing thecomputation of quantities of interest for a large number of agents with a much smallernumber of dynamical equations for averages for each type of agent with the help ofthe corresponding master equation.

The accuracy of the approximation nevertheless depends on avoiding oversimplifi-cations of the interactions between agents. In particular, as has been recently stressedin the literature on stock-flow consistent (SFC) models, economic agents are linkedby credit and debt relationships that put constraints on both individual and aggregatebalance sheets (see Caiani et al (2016) for a recent ABM-SFC model), and these inturn need to be taken into account in the mean-field approximation. We illustrate thisphenomenon in this paper using the models in Carvalho and Di Guilmi (2014) andDi Guilmi and Carvalho (2017) as our starting points.

In Section 2, we introduce an SFC ABM with two types of firms and two typesof households. The firms can be either aggressive (type 1) or conservative (type 2),depending on how much their current level of investment reacts to past profits. House-holds, on the other hand, can be either noninvestors (type 1) or investors (type 2),depending on whether or not they invest a portion of their savings in the stock mar-ket. In contrast with Di Guilmi et al (2010), Delli Gatti et al (2012), Carvalho andDi Guilmi (2014) and Di Guilmi and Carvalho (2017), in this paper we assume thatthe probabilities of transitions between types are constant and exogenously given. Asexplained in Appendix A, this is because, as far as we can tell, the solution method forthe master equation employed in those papers does not extend to the time-dependent,threshold-based transition probabilities they propose to use.2 By contrast, we showin Section 3 that, in the case of constant and exogenous transition probabilities, theso-called ansatz method, explained in detail in Aoki (2002) for the case of two typesof agents, extends to the 2 � 2 case: namely, when two types of agents in one sector(say, firms) interact with two types of agents in another sector (say, households).

Section 4 investigates both theABM and its mean-field approximation. We first ver-ify that the mean-field approximation gives rise to aggregate variables, such as equityprices and nominal output, that closely match the corresponding values obtained insimulations of the full ABM. Next we use the mean-field approximation to performexplorations of the parameter space that would be much slower withABM simulations.

2 As pointed out by an anonymous referee, an alternative solution method for the mean-fieldapproximation with time-varying transition rates has recently been proposed in Di Guilmi et al(2017).






In particular, we investigate the behavior of aggregate variables with respect to param-eters that are difficult to estimate outside the model, such as the fraction of externalfinancing that firms raise by issuing new debt as opposed to equity.

2 THE MODEL

We assume that the economy consists of an aggregate banking sector (henceforthreferred to as “the bank”), N firms indexed by n D 1; : : : ; N , and M householdsindexed by m D 1; : : : ; M . The N firms collectively produce a total output Qt at timet , which determines the total demand for labor and the total wage bill as Lt D Qt=a

and Wt D cQt , where a is the productivity per unit of labor and c is the labor costper unit of output. As in Carvalho and Di Guilmi (2014) and Di Guilmi and Carvalho(2017), we ignore labor market dynamics by assuming that both c and a are constant.Next, we assume that the price of each unit of output is given by

pt D �c; (2.1)

where � > 1 is constant markup over unit cost. It then follows that the wage share ofoutput is constant and is given by

! D Wt

pQt

D cQt

�cQt

D 1

�; (2.2)

and consequently the profit share of output is also constant and is given by

� D pQt � Wt

pQt

D 1 � ! D � � 1

�: (2.3)

We further assume that each household supplies Lt=M units of labor at time t ,thereby receiving a wage rate3

Wt D Wt

MD cQt

MD ca

Lt

M; (2.4)

which we assume to be the same for all households.4

3 Alternatively, we could follow Di Guilmi and Carvalho (2017) and assume that there are Lt

households employed at time t , each supplying one unit of labor at a constant average wage rateW D ca. The disadvantage of this approach is that the number of employed households fluctuatesin time, creating a distinction among households in addition to the types introduced in Section 2.2.4 In Carvalho and Di Guilmi (2014), each household is subject to a further idiosyncratic shock to itswage rate. We do not pursue this approach here, as the only sources of randomness in our model arethe transitions between types of firms and households with exogenous rates that were introduced inSection 2.3. Additional demand or supply shocks can be modeled separately.






2.1 Balance sheets

The balance sheets of each agent at time t are depicted in Table 1. Namely, firm n hasassets consisting of capital with nominal value pkn

t , it has liabilities consisting of netdebt with nominal value bn

t , and it has ent shares at an average price of p

ent , leading

to net worth equal tovn

t D pknt � bn

t � pent en

t : (2.5)

Note that, to simplify the notation, we treat net debt bnt as the difference between

loans and deposits for firm n, which can therefore be positive or negative dependingon whether firm n is a net borrower or lender (ie, depositor), respectively. Observethat we follow the accounting convention advocated in Godley and Lavoie (2007):namely, that equity issued by firms should be treated as a financial liability booked atmarket value. This is done for consistency with national accounts, where equity heldby households is treated as a financial asset for shareholders and is also accounted atmarket value. Note that the net worth in (2.5) resulting from this convention is typicallymuch smaller than the more common corporate accounting concept of shareholderequity, which in our context corresponds to

�n D pknt � bn

t ; (2.6)

that is, the “book value” of the difference between assets (physical capital anddeposits) and debt liabilities (loans). The discrepancy between the market and bookvalues of equity is captured by the valuation ratio or Tobin’s q, which in our contextreduces to

qnt D p

ent en

t C bnt

�n C bnt

D pknt � vn

t

pknt

; (2.7)

from which we can see that the net worth for firm n in (2.5) is positive if, and only if,its q-ratio is less than one, meaning that the market undervalues the firm.

Similarly, household m has assets consisting of emt shares at average price p

emt and

cash balances d mt deposited at the bank, leading to a net worth

vmt D p

emt em

t C d mt : (2.8)

Note that we again treat d mt as the difference between deposits and loans for household

m, which can therefore be positive or negative depending on whether household m is anet lender or borrower, respectively. The balance sheet of the bank accommodates thedemands for loans and deposits across the economy. Accordingly, its assets consistof aggregate net borrowing by firms,

Bt DNX

nD1

bnt ; (2.9)






TABLE 1 Balance sheets at t .

Firm n

bnt

pknt p

ent en

t

vnt

Household m

pemt em

tvm

tdm

t

BankBt Dt

Rt V bt

plus cash reserves Rt , and its liabilities consist of aggregate net deposits of households,

Dt DMX

mD1

d mt ; (2.10)

leading to a net worth of the form V bt D Bt C Rt � Dt .

Regarding equities, as we shall see below, we will assume a homogenous behaviorfor firms with respect to dividend payments and share issuance and buyback. Basedon this, we make the simplifying assumption that, instead of trading in shares forindividual companies, investors buy and sell shares of an aggregated fund at a com-mon price pe

t , which in turn buys and sells shares from firms. The price pet is then

determined by an equilibrium condition for the supply and demand for equities underthe constraint that

NXnD1

ent D Et D

MXmD1

emt : (2.11)

2.2 Transactions and aggregate demand

We consider a demand-driven economy operating below maximum capacity, so that,given aggregate demand Qt , firms adjust production according to

qnt D f n

t Qt ; f nt > 0;

NXnD1

f nt D 1; (2.12)

where the fractions f nt are known at time t . For example, we can have f n

t D f n0 for

a constant vector of allocations .f 10 ; : : : ; f n

0 /, or, alternatively, we could consider apreferential attachment rule of the form

f nt D kn

t

Kt

; Kt DNX

nD1

knt I (2.13)






that is to say, firms with larger capital at time t receive a larger share of demand.5 Eachfirm is classified as either aggressive (type 1) or conservative (type 2). We assume thatfirm n decides on its investment at t C 1 based on its previous type zn

t 2 f1; 2g, itsgross profits �pqn

t , its production level pqnt (used as a proxy for capacity utilization)

and its debt bnt according to

intC1 D ˛zn

t�pqn

t C ˇpqnt � �bn

t D .˛znt� C ˇ/pqn

t � �bnt ; (2.14)

where ˛z , ˇ and � denote the sensitivity of investment to gross profits, capacityutilization and current level of debt, respectively. We assume that ˛1 > ˛2; that isto say, investment by aggressive firms is more sensitive to gross profits than is thecase for conservative firms. This in turn determines capital for firm n at time t C 1

according topkn

tC1 D intC1 C .1 � ı/pkn

t ; (2.15)

as well as aggregate capital KtC1 DPN

nD1 kntC1. Observe, in particular, that the

aggregate capital evolves as

p.KtC1 � Kt / D ItC1 � ıpKt ; (2.16)

where ItC1 DPN

nD1 intC1 denotes total investment. Aggregate demand QtC1 is

determined by equilibrium in the goods market once consumption by households isspecified. Assuming that f n

tC1 is known at time t C 1 (a constant, say; or alternativelyequal to f n

tC1 D kntC1=KtC1), the share of production for firm n is again obtained as

qntC1 D f n

tC1QtC1; (2.17)

which in turn determines the gross profit for firm n as �pqntC1. The amount of retained

profits available to firm n to finance investment at time t C 1 is then given by

antC1 D �pqn

tC1 � rbnt � ıpkn

t � ıepet e

nt ; (2.18)

where rbnt are interest charges on debt held at time t , ıpkn

t are depreciation charges(otherwise known in accounting as consumption of fixed capital), and ıepe

t ent are

5 In Di Guilmi and Carvalho (2017), this share is further subject to an idiosyncratic shock thatredistributes demand among firms such that

EŒqnt � D kn

t

KtQt and

NXnD1

qnt D Qt :

We do not pursue this approach either, as the only source of randomness is the transition betweentypes of firms and households with exogenous rates introduced in Section 2.3. Additional demandor supply shocks can be modeled separately.






dividends paid to shareholders according to a dividend yield ıe, which we assume tobe constant and equal for all firms.

The two classes of households correspond to noninvestors (type 1), for whomem

t D 0, and investors (type 2), for whom emt > 0. Accordingly, the disposable

income to be received by household m at time t C 1 consists of

ymtC1 D WtC1 C rd m

t C ıepet e

mt ; (2.19)

where WtC1 D .1 � �/pQtC1=M is the effective wage rate obtained in (2.4), rd mt

is interest paid on deposits d mt held at time t , and the last term represents dividends

paid to household m, which we assume to be a fraction emt =Et of the total amount of

dividends ıepet Et paid by firms. In other words, we assume that household m receives

dividends in proportion to their equity holdings before rebalancing their portfolio, andin particular before changing type, at time t C 1.

Household m then decides on its consumption at time t C 1 based on its previousstate zm

t 2 f1; 2g, current disposable income ymtC1, and previous wealth vm

t D d mt C

pet e

mt according to

cmtC1 D .1 � s

y

zmt

/ymtC1 C .1 � sv

zmt

/vmt ; (2.20)

where syz ; sw

z 2 Œ0; 1� are the saving rates from income and wealth, respectively. Weassume that s

y1 6 s

y2 and sv

1 6 sv2 , so that investors save a higher proportion of both

income and wealth than noninvestors.Nominal aggregate demand at time t C 1 is then given by

pQtC1 D ItC1 C CtC1; (2.21)

with

ItC1 DNX

nD1

intC1 D �p.˛1Q1

t C ˛2Q2t / C ˇpQt � �Bt (2.22)

and

CtC1 DMX

mD1

cmtC1

D .1 � sy1 /Œ.1 � �/pQtC1m1

t C rD1t � C .1 � sv

1/D1t

C .1 � sy2 /Œ.1 � �/pQtC1m2

t C rD2t C ıepe

t Et �

C .1 � sv2/.D2

t C pet Et /; (2.23)

where we used the notation mzt for the proportion of households of type z at time t

and also introduced the class aggregates

Qzt D

Xfn W zn

t Dzgqn

t ; Dzt D

Xfm W zm

t Dzgd m

t : (2.24)






Substituting (2.22) and (2.23) into (2.21), we find that aggregate demand at t C 1 canbe calculated from quantities known at time t as follows:

pQtC1 D Ft

1 � .1 � �/Œ.1 � sy1 /m1

t C .1 � sy2 /m2

t �; (2.25)

where

Ft D �p.˛1Q1t C ˛2Q2

t / C ˇpQt � �Bt C .1 � sy1 /rD1

t C .1 � sv1/D1

t

C .1 � sy2 /.rD2

t C ıepet Et / C .1 � sv

2/.D2t C pe

t Et /: (2.26)

2.3 Transitions

We assume that, after making its investment decision for time t C 1, each firmn undergoes a transition to determine its new type according to the conditionalprobabilities

P fij .t/ WD Prob.zn

tC1 D j j znt D i/ D

1 � �f �f

�f 1 � �f

!: (2.27)

In words, each of the N 1t firms in state z D 1 (aggressive) at time t decides to

transition to state z D 2 (conservative) at time t C 1 with probability �f . Similarly,each of the N �N 1

t firms in state 2 at t decides to transition to state 1 with probability�f . Here, �f and �f are constant parameters specified exogenously.

Similarly, after making a consumption decision for time t C 1, each householdm undergoes a transition to determine its new type according to the conditionalprobabilities

P hij .t/ WD Prob.zm

tC1 D j j zmt D i/ D

1 � �h �h

�h 1 � �h

!: (2.28)

That is, each of the M 1t households in state z D 1 (noninvestor) at time t decides

to transition to state z D 2 (investor) at time t C 1 with probability �h. Similarly,each of the M � M 1

t households in state 2 at t decides to transition to state 1 withprobability �h.As with the rates for firms, �h and �h are constant parameters specifiedexogenously.

This specification of transition probabilities is a significant departure from themodels in Carvalho and Di Guilmi (2014) and Di Guilmi and Carvalho (2017), wherethe transition probabilities are specified as functions of balance sheet variables for eachagent. As a result, our model has both significantly different behavioral assumptionsand analytical properties.






More specifically, whereas we classify firms as either aggressive or conservativedepending on some exogenously determined propensity to invest, Di Guilmi andCarvalho (2017) classify them as speculative and hedge firms, depending on whetheror not they need to borrow in order to finance their investment. From a behavioralpoint of view, our assumption means that the balance sheet of firm n at time t affectsits investment decision only indirectly through the debt level bn

t in (2.14), whereasthe elasticity ˛zn

tdepends on an independently specified random variable: namely, its

type at time t , eg, having to do with the “animal spirits” of the managers of the firmat the time. In other words, in our model, two firms with identical balance sheets andfacing the same demand qt can still make different investment decisions according to(2.14) provided one is aggressive and the other is conservative at the time. By contrast,the type – and consequently the investment behavior – of a firm in Di Guilmi andCarvalho (2017) is entirely determined by its balance sheet: two firms with identicalbalance sheets and facing the same demand will necessarily make the same investmentdecision. Similar remarks apply to the classification of households in Carvalho andDi Guilmi (2014) as borrowing or nonborrowing, instead of investor or noninvestorin our model.

From an analytical point of view, as mentioned in Appendix A, the transition prob-abilities adopted in Carvalho and Di Guilmi (2014) and Di Guilmi and Carvalho(2017), which were themselves adapted from earlier work in Di Guilmi et al (2010),do not satisfy the conditions necessary to derive the approximation of the master equa-tion described in Section 3.2, making it difficult to justify the use of the mean-fieldapproximation for their models.

2.4 Flow of funds

When net investment intC1 � ıpkn

t for firm n exceeds its retained profits antC1, the

difference needs to be financed by new borrowing from the banking sector or issuanceof new shares. Conversely, if the amount of investment is lower than retained profits,then the excess funds can be used to pay down outstanding debt or to buy back shares.Following Carvalho and Di Guilmi (2014), we assume that firm n raises externalfunds according to the proportions

bntC1 � bn

t D $.intC1 � ıpkn

t � antC1/; (2.29)

petC1.en

tC1 � ent / D .1 � $/.in

tC1 � ıpknt � an

tC1/; (2.30)

where 0 6 $ 6 1 is a constant common to all firms. The debt bntC1 held by firm n

at time t C 1 can therefore be determined by (2.29), whereas the number of sharesen

tC1 outstanding for firm n at time t C 1 depends on the equity price petC1 in (2.31)






below. The total supply of equities at time t C 1 is given by

petC1EtC1 D pe

tC1Et C .1 � $/.ItC1 � ıpKt � AtC1/; (2.31)

where ItC1 is defined in (2.22) and

AtC1 DNX

nD1

antC1 D �pQtC1 � rBt � ıpKt � ıepe

t Et (2.32)

denotes the total amount of retained profits or, in other words, the total savings forthe firm sector.

On the other hand, savings for household m at time t C 1 are given by

smtC1 D ym

tC1 � cmtC1; (2.33)

which, as we have seen, only depends on quantities that are known at time t , includingits type zm

t and aggregate demand QtC1 given by (2.25). Accordingly, total savingsfor the household sector are given by

StC1 DMX

mD1

smtC1 D

MXmD1

.ymtC1 � cm

tC1/ D .1 � �/pQtC1 C rDt C ıepet Et � CtC1:

(2.34)The change in wealth for household m is then given by savings plus capital gains:

that is,

vmtC1 D vm

t C smtC1 C .pe

tC1 � pet /e

mt : (2.35)

This wealth at time t C 1 is then allocated into deposits and equities according to thenew type zm

tC1 for household m. Namely, we assume that the demand for equities forhousehold m is given by

petC1em

tC1 D 'vmtC1.zm

tC1 � 1/ D(

0 if zmtC1 D 1;

'vmtC1 if zm

tC1 D 2;(2.36)

where ' is a constant common to all households, and we recall that zmtC1 D 2 if

household m is an investor (type 2) and zmtC1 D 1 otherwise (type 1). The demand

for deposits for household m is then given by the residual

d mtC1 D vm

tC1 � petC1em

tC1 D(

vmtC1 if zm

tC1 D 1;

.1 � '/vmtC1 if zm

tC1 D 2:(2.37)






Accordingly, total demand for equities by households is given by

petC1EtC1 D '

� Xfm W zm

tC1D2g

vmtC1

�

D '

� Xfm W zm

tC1D2g

vmt C sm

tC1 C .petC1 � pe

t /emt

�

D '

� Xfm W zm

tC1D2g

d mt C pe

t emt C sm

tC1 C .petC1 � pe

t /emt

�

D '.D2;tC1t C S2

tC1 C petC1E

2;tC1t /; (2.38)

where we introduced the class aggregates

D2;tC1t D

Xfm W zm

tC1D2g

d mt ;

S2tC1 D

Xfm W zm

tC1D2g

smtC1;

E2;tC1t D

Xfm W zm

tC1D2g

emt :

In these expressions, note that the upper time index refers to the time in which thetype zm

tC1 is evaluated, whereas the lower time index refers to the time in which thesummands are evaluated. In words, D

2;tC1t is the sum of deposits held at time t by

households that are of type 2 at time t C 1. When the upper and lower time indexescoincide, we suppress the upper index, in accordance with the notation introducedin (2.24).

Equating the total supply of equities in (2.31) with the total demand for equities in(2.38) leads to an equilibrium equity price of the form

petC1 D

'.D2;tC1t C S2

tC1/ � .1 � $/.ItC1 � ıpKt � AtC1/

Et � 'E2;tC1t

: (2.39)

This can then be used in (2.30) to obtain the number of shares entC1 outstanding for

firm n at time t C 1, and consequently the total number of shares EtC1 DPN

nD1 entC1.

We can now perform two stock-flow consistency checks by calculating the savingsfor the bank at t C 1 as the change in its net worth: namely,

SbtC1 D V b

tC1 � V bt D .BtC1 � Bt / � .DtC1 � Dt /: (2.40)

Observe first that it follows from (2.29) that

BtC1 � Bt DNX

nD1

$.intC1 � ıpkn

t � antC1/ D $.ItC1 � ıpKt � AtC1/: (2.41)






Next, we compute the total amount of deposits at time t C 1 as

DtC1 DMX

mD1

d mtC1

DX

fm W zmtC1

D1gvm

tC1 CX

fm W zmtC1

D2g.1 � '/vm

tC1

DX

fm W zmtC1

D1gvm

t C smtC1 C .pe

tC1 � pet /e

mt

C .1 � '/

� Xfm W zm

tC1D2g

vmt C sm

tC1 C .petC1 � pe

t /emt

�

DX

fm W zmtC1

D1gd m

t C smtC1 C pe

tC1emt

C .1 � '/

� Xfm W zm

tC1D2g

d mt C sm

tC1 C petC1em

t

�

D D1;tC1t C S1

tC1 C petC1E

1;tC1t C .1 � '/.D

2;tC1t C S2

tC1 C petC1E

2;tC1t /

D Dt C StC1 C petC1Et � '.D

2;tC1t C S2

tC1 C petC1E

2;tC1t /

D Dt C StC1 C petC1Et � pe

tC1EtC1

D Dt C StC1 � .1 � $/.ItC1 � ıpKt � AtC1/; (2.42)

where we used vmt D d m

t C pet e

mt to move from the second to the third line above,

in addition to (2.38) and (2.31) in the last two lines. Substituting (2.41) and (2.42) in(2.40) gives

SbtC1 C StC1 C AtC1 D ItC1 � ıpKt ; (2.43)

which confirms that net investment at time t C 1 equals the total savings across thethree sectors in the economy. Further, using the expressions (2.32) and (2.34) we findthat

SbtC1 D ItC1 � ıpKt � .�pQtC1 � rBt � ıpKt � ıepe

t Et /

� Œ.1 � �/pQtC1 C rDt C ıepet Et � CtC1�

D ItC1 C CtC1 � pQtC1 C rBt � rDt D r.Bt � Dt /; (2.44)

confirming that profits for the bank consist of the interest differential between loansand deposits.

2.5 Special cases

When all households are of the same type z, aggregate disposable income becomes

YtC1 D .1 � �/pQtC1 C rDt C ıepet Et1fzD2g; (2.45)






where 1fzD2g D 1 indicates that all households are investors and 1fzD2g D 0 other-wise, and consumption is given by

CtC1 D .1 � syz /YtC1 C .1 � sv

z /Vt ; (2.46)

where Vt D Dt C pet Et1fzD2g. In this case, aggregate demand is given by

pQtC1 D ItC1 C .1 � syz /.rDt C ıepe

t Et1fzD2g/ C .1 � svz /Vt

1 � .1 � �/.1 � syz /

; (2.47)

where ItC1 is still given by (2.22). If all households are noninvestors, then (2.47)reduces to equation (19) in Di Guilmi and Carvalho (2017).6 In this case, we shouldalso impose that $ D 1, since there is no active equity market where firms can raisefunds. On the other hand, if all households are investors, then the equity price in (2.39)reduces to

petC1 D '.Dt C StC1/ � .1 � $/.ItC1 � ıpKt � AtC1/

.1 � '/Et

; (2.48)

which coincides with equation (35) in Carvalho and Di Guilmi (2014) with ' as aconstant proportion of wealth invested in equities instead of the variable proportionadopted in their equation (19).7

Conversely, when all firms are of the same type, aggregate investment becomes

ItC1 D .�˛ C ˇ/pQt � �Bt ; (2.49)

which reduces to equation (7) in Carvalho and Di Guilmi (2014) apart from obviousmodifications.8

3 MEAN-FIELD APPROXIMATION

The model of the previous section can be readily implemented as an ABM for reason-ably large numbers of firms and households. Because of the probabilistic nature of

6 With the extra assumption that households do not receive any interest on deposits, as is implicitlyassumed in Di Guilmi and Carvalho (2017).7 Note that our definition of retained profits At differs from that in Carvalho and Di Guilmi (2014)in two ways. First, we subtract depreciation costs from gross profits, as is commonly done inaccounting, while at the same time subtracting the same amount from gross investment. Second, weassume that distributed profits take the form of a constant dividend yield ıe, rather than a constantdividend payout ratio �, which avoids the anomaly of paying out negative dividends when earningsare negative.8 Namely, the effect of debt on investment is not considered in Carvalho and Di Guilmi (2014),corresponding to � D 0 in our setting. Conversely, we do not consider either a desired capacityutilization or the effect of stock valuation on investment, corresponding to setting their constants ud

and " to zero. Redefining the roles of ˛ and ˇ completes the identification between our equation (2.49)and their equation (7).






the transitions between types of agents, the effects of the different model parameterson the asymptotic properties of the model are not immediately clear, and algebraicmanipulation of the discrete-time equations governing its dynamic evolution provesto be both tedious and challenging. The purpose of this section is to present a mean-field approximation approach along the lines proposed in Di Guilmi et al (2010) andfollowed in Carvalho and Di Guilmi (2014) and Di Guilmi and Carvalho (2017), asan alternative to large-scale numerical simulations of the discrete-time ABM.

3.1 Discrete-time mean-field dynamics

The first ingredient of the approach consists in homogenizing the populations offirms and households of a given type by expressing the discrete-time model in termsof “mean-field” variables that are common to all agents of the same type. Note thatthis is very different from the representative-agent framework mentioned in Section 1in connection with DSGE models. Our use of mean-field values over a group ofagents of the same type is done purely for computational convenience, whereas theuse of representative agents is essentially the only way to guarantee the stability ofequilibrium in DSGE models as a consequence of the Sonnenschein–Mantel–Debreutheorem (see, for example, Kirman 1992).

We denote the average values of a variable x for agents of type z at time t by Nxzt .

Its time evolution requires two steps: we first compute the deterministic value QxztC1

before agents change type at time t C 1 and then calculate the new mean-field valueNxz

tC1 taking into account the changes in type. This is necessary because agents carrytheir balance sheet items with them when they change type, and consequently both theaggregate and average values of a variable for agents of type z change when agentschange type.9 Accordingly, since the average number of firms changing from type 1to type 2 is �N 1

t and the average number of firms changing from type 2 to type 1 is�.N � N 1

t /, we set the mean-field values of x for firms of type z after a change oftype at time t C 1 to be

Nx1tC1 D

.1 � �/N 1t Qx1

tC1 C �.N � N 1t / Qx2

tC1

N 1tC1

(3.1)

and

Nx2tC1 D

�N 1t Qx1

tC1 C .1 � �/.N � N 1t / Qx2

tC1

N � N 1tC1

: (3.2)

9 Rebalancing after a change of type does not seem to be considered in either Carvalho and Di Guilmi(2014) or Di Guilmi and Carvalho (2017), even though this leads to puzzling behavior in aggregatevariables. For example, letting p Nkz

tC1 D NiztC1 C .1 � ı/p Nkz

t and ignoring rebalancing, it is easyto see that pKtC1 ¤ ItC1 C .1 � ı/pKt , where KtC1 D N 1

tC1Nk1tC1 C .N � N 1

tC1/ Nk2tC1,

ItC1 D N 1tC1

Ni1tC1 C .N � N 1

tC1/Ni2tC1 and Kt D N 1

tNk1t C .N � N 1

t / Nk2t .






In this way, we find that

XtC1 D N 1tC1 Nx1

tC1C.N �N 1tC1/ Nx2

tC1 D N 1t Qx1

tC1C.N �N 1t / Qx2

tC1 D QXtC1; (3.3)

so the aggregate values for the variable x across the entire economy are the same beforeand after a change of type at time t C 1, as they should be. Similar expressions holdfor mean-field variables for households, with M z

t and M ztC1 replacing N z

t and N ztC1.

In the context of the present model, suppose that, at time t , we are given the totalproduction Qt , the mean-field variables Nkz

t , Nbzt , Nez

t , and the number of firms N zt of

type z D 1; 2, as well as the mean-field variable Nd zt and the number of households

M zt of type z D 1; 2. We then compute the mean-field production for each type as

the analogs of (2.12): that is,

Nqzt D Nf z

t Qt ; z D 1; 2; (3.4)

where Nf zt > 0 are known at time t and satisfy N 1

tNf 1t C .N � N 1

t / Nf 2t D 1. For

example, they can be set to Nf zt D Nf 1

0 =N 1t for a constant vector . Nf 1

0 ; Nf 20 / satisfying

Nf 10 C Nf 2

0 D 1 or, alternatively, they can be set to be proportional to the mean-fieldcapital for each type: that is, Nf z

t D Nkzt =Kt , where Kt D N 1

tNk1t C .N � N 1

t / Nk2t is

the aggregate capital. The mean-field investment demand before firms change type att C 1 is then given by the analog of (2.14): namely,

QiztC1 D .˛z� C ˇ/p Nqz

t � � Nbzt ; z D 1; 2: (3.5)

Accordingly, the mean-field capital QkztC1 before a change in type at t C 1 is given by

the analog of (2.15): that is,

p QkztC1 D Qiz

tC1 C .1 � ı/p Nkzt ; z D 1; 2: (3.6)

Once aggregate demand QQtC1 is determined by equilibrium in the goods market, themean-field productions before firms change type at t C 1 can then be obtained as

QqztC1 D Qf z

tC1QQtC1; (3.7)

where Qf ztC1 is known at t C 1 before firms change type (it is given, say, by Qf z

tC1 DQkztC1= QKtC1, where QKtC1 D N 1

tQk1tC1 C .N � N 1

t / Qk2tC1 denotes aggregate capital

in the economy before firms change type at t C 1). After paying r Nbzt as interest

charges on mean-field debt, ıp Nkzt as depreciation costs for the mean-field capital,

and dividends ıepet Nez

t , all based on holdings at time t , the mean-field retained profitsbefore changing type at t C 1 are calculated as

QaztC1 D �p Qqz

tC1 � r Nbzt � ıp Nkz

t � ıepet Nez

t ; (3.8)






so that aggregate retained profits are given by

QAtC1 D �p QQtC1 � rBt � ıpKt � ıepet Et : (3.9)

As in theABM model, net investment in excess of retained profits needs to be financedexternally by new debt and share issuance as follows:

QbztC1 � Nbz

t D $.QiztC1 � ıp Nkz

t � QaztC1/; (3.10)

petC1. Qez

tC1 � Nezt / D .1 � $/.Qiz

tC1 � ıp Nkzt � Qaz

tC1/: (3.11)

We therefore have that the total supply of equities offered by firms satisfies

petC1

QEtC1 D .1 � $/. QItC1 � ıp QKt � QAtC1/ C petC1

QEt : (3.12)

Moving to households, the mean-field disposable incomes for types z D 1; 2 beforea change in type at time t C 1 are given by

Qy1tC1 D WtC1 C r Nd 1

t ; (3.13)

Qy2tC1 D WtC1 C r Nd 2

t C ıepet

Et

.M � M 1t /

; (3.14)

where WtC1 D .1 � �/p QQtC1=M . In other words, an equal fraction .M � M 1t /�1

of distributed profits ıepet Et is paid to each of the .M � M 1

t / households of type 2before a change of type at time t C 1. Here, QQtC1 D N 1

t Qq1tC1 C .N � N 1

t / Qq2tC1.

The mean-field consumptions before a change in type at time t C 1 are then

Qc1tC1 D .1 � s

y1 /.WtC1 C r Nd 1

t / C .1 � sv1/ Nd 1

t ; (3.15)

Qc2tC1 D .1 � s

y2 /

�WtC1 C r Nd 2

t C ıepet

Et

.M � M 1t /

�

C .1 � sv2/

�Nd 2t C pe

t

Et

.M � M 1t /

�: (3.16)

We therefore have that aggregate demand at time t C 1 before the change of typefor firms and households is given by

p QQtC1 D QItC1 C QCtC1; (3.17)

with

QItC1 D N 1t

Qi1tC1 C .N � N 1

t /Qi2tC1 D �p.˛1Q1

t C ˛2Q2t / C ˇpQt � �Bt (3.18)

and

QCtC1 D M 1t Qc1

tC1 C .M � M 1t / Qc2

tC1

D .1 � �/p QQtC1

MŒ.1 � s

y1 /M 1

t C .1 � sy2 /.M � M 1

t /� C .1 � sy1 /rD1

t

C .1 � sy2 /.rD2


1/D1t C .1 � sv

2/.D2t C pe

t Et /;

(3.19)






where we used the aggregate variables

Qzt D N z

t Nqzt ; Dz

t D M zt

Nd zt : (3.20)

Substituting (3.19) into (3.17), we find that aggregate demand before a change of typeat t C 1 can be calculated from quantities known at time t as follows:

p QQtC1 D Ft

1 � .1 � �/Œ.1 � sy1 /m1

t C .1 � sy2 /m2

t �; (3.21)

where

Ft D �p.˛1Q1t C ˛2Q2

t / C ˇpQt � �Bt C .1 � sy1 /rD1

t

C .1 � sy2 /.rD2


1/D1t C .1 � sv

2/.D2t C pe

t Et /:

(3.22)

The mean-field savings for the two types of households before changing type attime t C 1 are then given by

QsztC1 D Qyz

tC1 � QcztC1; (3.23)

which, as in the ABM model, only depends on quantities known at time t . Totalsavings for the household sector before a change of type at t C 1 are then given by

QStC1 D M 1t Qs1

tC1C.M �M 1t /Qs2

tC1 D .1��/p QQtC1CrDt Cıepet Et � QCtC1: (3.24)

The mean-field wealth values for the two types of households before changing typeat t C 1 are given by wealth at time t , plus savings, plus capital gains: that is,

Qv1tC1 D Nv1

t C Qs1tC1; (3.25)

Qv2tC1 D Nv2

t C Qs2tC1 C .pe

tC1 � pet /

Et

.M � M 1t /

: (3.26)

Having computed QvztC1 before a change of type, we let households change type at

time t C 1 as described in Section 3.2 and calculate the new mean-field values NvztC1

according to the expressions (3.1) and (3.2) (suitably modified for M zt instead of

N zt ). The wealth for each type of agent is then reallocated into deposits and equities

according to the new type at time t C 1 as follows:

Nd 1tC1 D Nv1

tC1; (3.27)Nd 2tC1 D .1 � '/ Nv2

tC1; (3.28)

petC1

EtC1

.M � M 1tC1/

D ' Nv2tC1: (3.29)






Accordingly, total demand for equities by households is given by

petC1EtC1 D '.M � M 1

tC1/ Nv2tC1 D '.D

2;tC1t C S2

tC1 C petC1E

2;tC1t /; (3.30)

where the mean-field analogs of the class aggregates introduced in (2.38) are

D2;tC1t D �M 1

tNd 1t C .1 � �/.M � M 1

t / Nd 2t ;

S2tC1 D �M 1

t Qs1tC1 C .1 � �/.M � M 1

t /Qs2tC1;

E2;tC1t D .1 � �/Et :

The interpretation of the upper and lower indexes here is the same as before. Forexample, D

2;tC1t corresponds to deposits held at time t by households that are of

type 2 at time t C 1. Equating the total supply of equities (3.12) with the total demand(3.30), we find the equilibrium equity price at time t C 1 is given by

petC1 D

'.D2;tC1t C S2

tC1/ � .1 � $/.ItC1 � ıpKt � AtC1/

Œ1 � '.1 � �/�Et

; (3.31)

from which we can calculate QeztC1 in (3.11). Finally, having computed Qkz

tC1, QeztC1 and

QbztC1, we calculate the new mean-field values Nkz

tC1, NbztC1 and Nez

tC1 according to theexpressions (3.1) and (3.2), and we verify that

p.KtC1 � Kt / D ItC1 � ıpKt ; (3.32)

BtC1 � Bt D $.ItC1 � ıpKt � AtC1/; (3.33)

petC1.EtC1 � Et / D .1 � $/.ItC1 � ıpKt � AtC1/: (3.34)

The same stock-flow consistency checks that we performed for the ABM model cannow be done here. Observe that

DtC1 D M 1tC1

Nd 1tC1 C .M � M 1

tC1/ Nd 2tC1

D M 1tC1 Nv1

tC1 C .1 � '/.M � M 1tC1/ Nv2

tC1

D .1 � �/M 1t Qv1

tC1 C �.M � M 1t / Qv2

tC1

C Œ�M 1t Qv1

tC1 C .1 � �/.M � M 1t / Qv2

tC1� � '.M � M 1tC1/ Nv2

tC1

D M 1t . Nd 1

t C Qs1tC1/ C .M � M 1

t /

�Nd 2t C Qs2

tC1 C petC1

Et

.M � M 1t /

�

� '.M � M 1tC1/ Nv2

tC1

D M 1t . Nd 1

t C Qs1tC1/ C .M � M 1

t /

�Nd 2t C Qs2

tC1

�C pe

tC1Et

� '.D2;tC1t C S2

tC1 C petC1E

2;tC1t /

D Dt C QStC1 C petC1Et � pe

tC1EtC1;






where we have used (3.30). Using (3.34), we conclude that

DtC1 � Dt D QStC1 � .1 � $/.ItC1 � ıpKt � AtC1/: (3.35)

Using this and (3.33), we find that

SbtC1 C QStC1 C AtC1 D .BtC1 � Bt / � .DtC1 � Dt / C StC1 C AtC1

D ItC1 � ıpKt ; (3.36)

so total savings across all sectors of the economy equals net investment. Finally,inserting (3.17), (3.24) and (3.9) into (3.36) and using the fact that, by construction,XtC1 D QXtC1 for aggregate quantities, we find that

SbtC1 D ItC1 � ıpKt � Œ.1 � �/p QQtC1 C rDt C ıepe

t Et � QCtC1�

� .�p QQtC1 � rBt � ıpKt � ıepet Et /

D ItC1 C CtC1 � pQtC1 C rBt � rDt D r.Bt � Dt /; (3.37)

so, as before, profits for the bank are accrued from the interest differential betweenloans and deposits.

3.2 Approximate continuous-time Markov chain dynamics

The second ingredient of the approach consists in approximating the discrete-timetransitions between types by a two-dimensional continuous-time Markov chain withstate .N 1

t ; M 1t /, that is, the numbers of aggressive firms and noninvestor households

at time t , and state space f0; 1; : : : ; N g � f0; 1; : : : ; M g. Accordingly, we assumethat the Markov chain at state .n; m/ can jump to one of four neighboring states.n ˙ 1; m ˙ 1/ with transition rates given by

d f.n/ D �fn; bf.n/ D �f.N � n/;

d h.m/ D �hm; bh.m/ D �h.M � m/:

)(3.38)

In other words, a jump from n to n � 1, corresponding to the “death” of a type 1firm, occurs in a small time interval dt with probability d f.n/dt , obtained as theprobability of an individual firm making the transition from type 1 to type 2, whichis given by �f according to (2.27), multiplied by the number n of firms currently oftype 1. Similarly, a jump from n to nC1, corresponding to the “birth” of a type 1 firm,occurs in an small time interval dt with probability bf.n/dt , obtained as the probability�f of an individual firm making the transition from type 2 to type 1 multiplied by the






number .N � n/ of firms currently of type 2. The death and birth transition rates forhouseholds are obtained analogously. Observe that these calculations for transitionrates assume that the change in type for different firms and households are independentrandom events, hence the multiplication of each individual transition probability bythe number of agents undergoing that transition.

The third and final ingredient consists in approximating the solution of the masterequation: namely, the equation governing the time evolution of the probability

P.n; mI t / D Prob.N 1t D n; M 1

t D m/: (3.39)

As shown in Appendix A, assuming that the numbers of firms and households oftype 1 at time t can be written as

N 1t D N�f.t/ C

pN � f.t/; M 1

t D M�h.t/ Cp

M �h.t/ (3.40)

for determinist functions �f.t/ and �f.t/, corresponding to their trends and stochasticprocesses � f.t/ and �h.t/ for random fluctuations around the trend, we obtain thefollowing ordinary differential equations:

d�f

dtD �f � .�f C �f/�f ;

d�h

dtD �h � .�h C �h/�h; (3.41)

from which it is easy to see that

�f.t/ D �f

�f C �fC e�.�f C�f /t

��f.0/ � �f

�f C �f

�

) �f1 WD lim

t!1�f.t/ D �f

�f C �f; (3.42)

�h.t/ D �h

�h C �hC e�.�hC�h/t

��h.0/ � �h

�h C �h

�

) �h1 WD lim

t!1�h.t/ D �h

�h C �h: (3.43)

Moreover, the probability densities of the random fluctuations satisfy two associatedFokker–Planck equations of the form (A.21), from which it follows that the fluctua-tions are asymptotically Gaussian distributed with means equal to zero and variancesgiven by

�2f D �f�f

.�f C �f/2; �2

h D �h�h

.�h C �h/2: (3.44)






The fractions n1t D N 1

t =N and m1t D M 1

t =M can therefore be approximated bystochastic differential equations of the form

dn1t D .�f C �f/

��f

�f C �f� n1

t

�dt C

s2�f�f

N.�f C �f/dW f

t ; (3.45)

dm1t D .�f C �f/

��h

�h C �h� m1

t

�dt C

s2�h�h

N.�h C �h/dW h

t (3.46)

for independent Brownian motions .W ft ; W h

t /.In summary, the mean-field model consists of the deterministic evolutions (3.6),

(3.10)–(3.11), (3.27)–(3.28) for the eight state variables . Nkz; Nbz; Nez; Nd z/, with z D1; 2, coupled with the stochastic evolution (3.45)–(3.46) for the fractions of firmsand households of type 1 (with the corresponding rebalancing after each change intype according to expressions (3.1) and (3.2)). In other words, the mean-field modelcorresponds to a ten-dimensional random dynamical system. By comparison, the fullABM requires the calculation of four state variables .kn

t ; bnt ; en

t ; znt / for each firm and

three state variables .d mt ; em

t ; zmt / for each household.

4 NUMERICAL SIMULATIONS

In this section, we illustrate the properties of the model by simulating both thefull ABM and the mean-field approximation using the base parameters describedin Table 2. The parameters were chosen consistently with the assumption that thediscrete-time equations in the model correspond to quarterly updates, that is, thebasic time period in the model is 0.25 years. In particular, the one-period depreci-ation rate ı, the dividend yield ıe and the interest rate r were chosen consistentlywith annualized rates of 4% for each variable. For the agent-based simulations, wetake p � 1:4, pe

0 D 1 and initialize the aggregate balance sheet items for firms atpK0 D 1400, B0 D 667, E0 D 333, leading to the initial aggregate net worth of thefirm sector being equal to V F

0 D 400, and the aggregate balance sheet items for thehousehold sector at D0 D 1067 and pe

0E0 D 333, leading to the initial aggregate networth of the household sector being equal to V H

0 D 1400.10 We then assume that theseaggregate amounts are uniformly distributed among individual firms and households,respectively.

We first compare the number of firms and households of each type obtained fromthe agent-based simulation and the mean-field approximation in Figure 1. Next, inFigure 2 we compare the time evolution for equity prices and nominal output obtained

10 We also assume a constant level of cash reserves for the bank R0 D 400, so that the initial networth of the bank implied by the aggregate balance sheets of firms and households is V B

0 D 0.






TABLE 2 Baseline parameter values.

Symbol Value Description

N 1000 Number of firms

M 4000 Number of households

a 1 Labor productivity

c 1 Unit labor cost

� 1.4 Markup factor

˛1 0.575 Profit elasticity of investment for aggressive firms

˛2 0.4 Profit elasticity of investment for conservative firms

ˇ 0.16 Utilization elasticity of investment

� 0.05 Debt elasticity of investment

r 0.01 One-period interest rate on loans and deposits

ı 0.01 One-period depreciation rate

ıe 0.01 One-period dividend yield

sy1 0.15 Propensity to save from income for noninvestors

sy2 0.4 Propensity to save from income for investors

sv1 0.85 Propensity to save from wealth for noninvestors

sv2 0.85 Propensity to save out of wealth for investors

�f 0.6 Transition probability from aggressive toconservative type for firms

�f 0.4 Transition probability from conservative toaggressive type for firms

�h 0.2 Transition probability from noninvestors toinvestors type for households

�h 0.3 Transition probability from investors to noninvestorstype for households

$ 0.6 Proportion of external financing for firms obtainedissuing new debt

' 0.5 Proportion of investor household wealth allocatedto stocks

from each method. We can observe a close match between the computationally inten-sive ABM and its mean-field approximation, both in terms of population fractions ofeach type of agent and in the resulting aggregate variables represented by the equityprices and output.

Next, in Figures 3–8 we use the mean-field approximation to perform a seriesof sensitivity tests with respect to several discretionary parameters. Starting withFigure 3, we see that, as expected, the return on equity decreases linearly with thedividend yield ıe.We also see that the growth rate of output increases with the dividend






FIGURE 1 Number of firms and households of each type obtained through simulationsof the agent-based model (ABM) and mean-field (MF) approximations.

0 100 200 300 400 500Period

(a)

0 100 200 300 400 500Period

(b)0.8

Per

cent

age

0.7

0.6

0.5

0.3

0.2

0.4

0.8

Per

cent

age

0.7

0.6

0.5

0.3

0.2

0.4

ABM aggressive firmsABM conservative firmsMF aggressive firmsMF conservative firms

ABM noninvestorABM investorMF noninvestorMF investor

(a) Proportions of aggressive and conservative firms. The average fraction of type 1 (aggressive) firms is 0.4.(b) Proportions of noninvestor and investor households. The average fraction of type 1 (noninvestor) households is0.6.

yield. This happens because, in our model, an increase in divided yield leads to higherdisposable income of households and, consequently, higher consumption, whereas theoffsetting decrease in aggregate investment is less pronounced, as firms can borrowthe necessary amount to finance investment. The base value ıe D 0:01, correspondingto an annual dividend yield of 4%, is compatible with average observed yields andleads to an average 2.7% growth rate in equity and an average 3.0% growth rate ofnominal output in our model.






FIGURE 2 Comparison between aggregate variables in the agent-based model (ABM)and in the mean-field (MF) approximation.

25

Pric

e

20

15

10

5

00 100 200 300 400 500

Period

(a)

ABM equity priceMF equity price

60 000

Out

put

50 000

40 000

30 000

10 000

00 100 200 300 400 500

Period

(b)

ABM outputMF output

20 000

(a) Equilibrium equity price. The initial equity price is pe0 D 1; the average annual return over the 120-year period

is 2.7% for both the ABM simulation and the mean-field approximation. (b) Aggregate output. The initial output isQ0 D 1000; the average annual growth rate over the 120-year period is 2.9% for both the ABM simulation and themean-field approximation.

The sensitivity test for the proportion $ of external financing that firms raisedthrough new debt is shown in Figure 4 and confirms that the base value chosen forthis parameter lies in a range where aggregate variables such as equity prices andoutput are not only realistic but relatively stable with respect to small changes in theparameters. The results in Figure 4 suggest that the value of a firm – reflected here bythe equilibrium equity price – is independent of the particular mix of debt and equity






FIGURE 3 Sensitivity of equity price and aggregate output to the dividend yield ıe (recallthat the annualized dividend yield is given by 4 � ıe).

0.030

0.025

0.020

0.015

0.010

Mea

n

0.00720

0.00715

0.00710

0.00705

0.00700

SD

0.0250 0.050 0.075 0.100 0.125 0.150 0.175 0.200

0.0250 0.050 0.075 0.100 0.125 0.150 0.175 0.2004 × de

(a)

0.0330

0.0325

0.0320

0.0315

0.0310

Mea

n

0.0070

0.0065

0.0060

0.0055

SD

0.0250 0.050 0.075 0.100 0.125 0.150 0.175 0.200

0.0250 0.050 0.075 0.100 0.125 0.150 0.175 0.200

4 × de

(b)

(a) Mean and standard deviation (SD) for the growth rate of equilibrium equity price. (b) Mean and standard deviation(SD) for the growth rate of aggregate output.






FIGURE 4 Sensitivity of equity price and aggregate output to the proportion $ of externalfinancing raised by debt.

0.030

0.025

0.020

0.015

0.010

Mea

n

0.025

0.020

0.015

0.010

SD

0.20 0.4 0.6 0.8 1.0

(a)

(b)

0.20 0.4 0.6 0.8 1.0ϖ

0.0314

0.0312

0.0310

0.0308

0.0306

Mea

n

0.0072

0.0071

0.0070

0.0069

SD

0.20 0.4 0.6 0.8 1.0

0.20 0.4 0.6 0.8 1.0ϖ

0.0304







FIGURE 5 Sensitivity of equity price and aggregate output to the proportion ' of householdwealth invested in stocks.

0.034

0.032

0.030

0.028

0.024

Mea

n

0.05

0.04

0.03

0.01

SD

0.30.1 0.5 0.7 0.9 1.0

(a)

ϕ

0.026

0.02

0

0.2 0.4 0.6 0.8

0.30.1 0.5 0.7 0.9 1.00.2 0.4 0.6 0.8

0.035

0.034

0.033

0.032

0.030

Mea

n

0.010

0.009

0.008

0.006

SD

0.30.1 0.5 0.7 0.9 1.0

(b)

ϕ

0.031

0.007

0.005

0.2 0.4 0.6 0.8

0.30.1 0.5 0.7 0.9 1.00.2 0.4 0.6 0.8







FIGURE 6 Sensitivity of equity price, aggregate output and debt-to-output ratio to theprofit elasticity of investment ˛ for firms. [Figure continues on next page.]

0.2

0

–0.4

–0.6

Mea

n

0.011

0.010

0.008SD

0.30.1 0.5 0.7 0.9

(a)

α

0.006

0.2 0.4 0.6 0.8

0.30.1 0.5 0.7 0.90.2 0.4 0.6 0.8

–0.2

0.009

0.007

(b)

0.300.25

0.15

0.05

Mea

n

0.020

0.015

0.010SD

0.30.1 0.5 0.7 0.9

α

0.10

0.005

0.2 0.4 0.6 0.8

0.30.1 0.5 0.7 0.90.2 0.4 0.6 0.8

0.20







FIGURE 6 Continued.

(c)

0.5

0

–1.0

–1.5

Mea

n

0.5

0.4

0.2

SD

0.30.1 0.5 0.7 0.9

α

0

0.2 0.4 0.6 0.8

0.30.1 0.5 0.7 0.90.2 0.4 0.6 0.8

–0.5

0.3

0.1

(c) Mean and standard deviation (SD) for the debt-to-output ratio.

used to finance its operations. That this seems to break down for $ < 0:4 is puzzlingand merits further investigation. Figure 5 shows a similar result for the parameter ',where it is interesting to see that the volatility of equity prices tends to increase bothwhen investors put all their wealth in stocks (ie, when ' ! 1) and when none of theirwealth is in stocks (ie, when ' ! 0).

Figure 6 shows the sensitivity tests for the profit elasticity parameter, where for thepurposes of the test we took ˛ D ˛1 D ˛2: that is to say, we assumed it was equalfor all firms. As we can observe in part (b) of the figure, the growth rate of outputincreases with ˛, since a higher value for this parameter leads to higher investmentby firms. On the other hand, as part (a) illustrates, an increasing value of ˛ has anegative effect on the growth rate of equity prices, as firms need to raise more fundsfor external financing and therefore increase the supply of equities. For the samereason, higher values of ˛ lead to higher debt-to-output ratios, as firms also need toborrow more to finance investment. Our base parameters reflect a choice where thelevel of responsiveness of investment to past profits is high enough to promote growthbut not so high as to compromise the financial viability of firms.

Figure 7 shows similar results for the saving rate from income sy , which we assumedto be the same for all households for the purpose of the sensitivity tests. As expected,






FIGURE 7 Sensitivity of equity price and aggregate output to the savings rate from incomesy for households.

0.035

0.030

0.025

0.020

0.010

Mea

n

0.014

0.012

0.010

0.016

SD

0.30.1 0.5 0.7 0.9

(a)

sy

0.015

0.008

0.2 0.4 0.6 0.8

0.30.1 0.5 0.7 0.90.2 0.4 0.6 0.8

0.040

0.035

0.030

0.025

0.020

Mea

nS

D

0.30.1 0.5 0.7 0.9

(b)

sy

0.2 0.4 0.6 0.8

0.30.1 0.5 0.7 0.90.2 0.4 0.6 0.8

0.030

0.025

0.020

0.010

0.015

0.005







FIGURE 8 Sensitivity of equity price and aggregate output to the savings rate from wealthsv for households.

0.030.020.01

0

–0.02

Mea

n

0.014

0.012

0.010

0.006

SD

0.30.1 0.5 0.7 0.9

(a)

sv

–0.01

0.008

0.2 0.4 0.6 0.8

0.30.1 0.5 0.7 0.90.2 0.4 0.6 0.8

0.055

0.050

0.045

0.040

0.030

Mea

n

0.06

0.04SD

0.30.1 0.5 0.7 0.9

(b)

sv

0.035

0.02

0.2 0.4 0.6 0.8

0.30.1 0.5 0.7 0.90.2 0.4 0.6 0.8

–0.03







we see in part (b) that aggregate output decreases with savings from income, as thisshifts household spending from consumption to accumulation of bank deposits andstocks, thereby raising the growth rate of equity as shown in part (a) of the samefigure. This effect is all the more pronounced when we consider the saving rate fromwealth sv in Figure 8. As we can see, a high propensity to spend accumulated wealth(correspondingly low sv) leads to a high growth rate but disastrous equity prices(both volatile and with negative returns). Conversely, total reinvestment of wealth (ie,sv ! 1) leads to high returns (but also high volatility) in stock prices but precipitouslylow growth for the economy as a whole. Our baseline parameters reflect a compromisebetween these two conflicting tendencies.

5 CONCLUSION AND FURTHER WORK

We have proposed a mean-field approximation to a stock-flow consistent ABM withheterogeneous firms and households. The approximation is inspired by earlier workin Di Guilmi et al (2010) and Delli Gatti et al (2012), but it differs from thesepapers in two fundamental respects. First, we take the transition rates between typesto be exogenous and constant, as this is the case for which the solution methodfor the master equation described in Appendix A applies. Second, we introduce anadditional rebalancing of mean-field variables (namely, (3.1) and (3.2)) that is imposedby stock-flow consistency and seems to have been previously neglected.

Our model for different firms is motivated by Di Guilmi and Carvalho (2017), exceptthat we classify firms into aggressive and conservative, rather than self-financing andnon-self-financing. In other words, the amount a firm decides to invest depends on aninherent property (eg, the “animal spirits” of its managers) rather than on its financialposition, which is then determined afterward depending on the overall state of theeconomy. Similarly, our model for different households is motivated by Carvalho andDi Guilmi (2014), except that we classify households into noninvestors and investors,rather than borrowers and nonborrowers. In other words, a household’s decision toinvest on the stock market depends on an inherent property (eg, the degree of riskaversion) rather than on its financial position.

With these two modifications, we obtain remarkable accuracy in the mean-fieldapproximation of aggregate variables when compared with the simulations of theunderlying ABM. We then use the mean-field approximation to perform a series ofsensitivity tests for the model with respect to some of its parameters, notably thedividend rate ıe, the proportion $ of external financing that firms raise from newdebt, the proportion ' of household wealth invested in the stock market, the elasticity˛ of investment to profits, and the propensity sy for households to save from income.These tests allow us to investigate the range of parameters that result in plausiblebehavior for the aggregate variables in the model.






For example, a sufficiently high value for the fraction $ of external finance raisedby issuing debt or the fraction ' of household wealth invested in equities leads to stableequity prices, characterized by high return and low volatility, as shown in Figure 2.Accordingly, we can simulate models with more turbulent stock markets – that is tosay, markets characterized by crashes and periods of high volatility – by loweringthe values of these parameters. A natural follow up question, motivated by Minsky’sfinancial instability hypothesis (see Minsky 1982), is whether a suitable extension ofthe model could allow for a stable scenario to evolve into an unstable one.

One way to achieve this is to introduce more interactions between the agents thanwe considered in this paper. Specifically, we could let the death and birth probabilitiesfor firms in Section 3.2 take the form

d f.n/ D �f�1

�n

N

�n; bf.n/ D �f�2

�n

N

�.N � n/ (5.1)

for functions �1.�/ and �2.�/ related to the relative gains from being of one type ratherthan another, and the solution method presented in Aoki (2002) still applies to thistype of transition probability. For example, the functions �1.�/ and �2.�/ can be relatedto profits for firms of different types, so that higher profits for aggressive firms wouldlead to more firms becoming aggressive, and it is plausible to conjecture that this kindof endogenous transition probability can generate instability from periods of stability,but we defer this investigation to future work.

APPENDIX A. APPROXIMATE SOLUTION TO THE MASTEREQUATION

We adjust the solution method used in Di Guilmi et al (2010), which is itself adaptedfrom Aoki (2002) and earlier references, to the case where there are two types of firmsand two types of households. Let .N 1

t ; M 1t / 2 f0; 1; : : : ; N g � f0; 1; : : : ; M g denote

the number of firms of type 1 and the number of households of type 1, respectively.It follows from the Markov property that the joint probability

P.n; mI t / D Prob.N 1t D n; M 1

t D m/ (A.1)

satisfies the so-called master equation: that is,

@P.n; mI t /

@tD d f.n C 1/P.n C 1; mI t / C bf.n � 1/P.n � 1; mI t /

C d h.m C 1/P.n; m C 1I t / C bh.m � 1/P.n; m � 1I t /

� Œd f.n/ C bf.n/ C d h.m/ C bh.m/�P.n; mI t /; (A.2)

with the obvious modifications at the boundaries n D m D 0, n D N and m D M .Here, the “death” and “birth” transition probabilities are defined in (3.38). Assuming






that firms and households choose their type independently from each other, we havethat

P.n; mI t / D P.n; t/P.m; t/; (A.3)

where P.n; t/ D Prob.N 1t D n/ and P.m; t/ D Prob.M 1

t D m/. Substituting (A.3)on both sides of (A.2) leads to

@P.n; t/

@tP.m; t/ C P.n; t/

@P.m; t/

@t

D .d f.n C 1/P.n C 1; t/ C bf.n � 1/P.n � 1; t//P.m; t/

C .d h.m C 1/P.m C 1; t/ C bh.m � 1/P.m � 1; t//P.n; t/

� Œd f.n/ C bf.n/�P.n; t/P.m; t/ � Œd h.m/ C bh.m/�P.n; t/P.m; t/:

(A.4)

Assuming further that P.n; t/ ¤ 0 and P.m; t/ ¤ 0 for all n, m, we find that (A.4)decouples into the following equations:

@P.n; t/

@tD d f.n C 1/P.n C 1; t/ C bf.n � 1/P.n � 1; t/

� Œd f.n/ C bf.n/�P.n; t/; (A.5)

@P.m; t/

@tD d h.m/P.m C 1; t/ C bh.m � 1/P.m � 1; t/

� Œd h.m/ C bh.m/�P.m; t/; (A.6)

which are identical to the master equation analyzed in Di Guilmi et al (2010). Weproceed with the analysis in terms of firms, with the results for households followingfrom obvious modifications. As in Di Guilmi et al (2010), for a generic function a.n/

define the lead and lag operators as

LŒa.n/� D a.n C 1/; L�1Œa.n/� D a.n � 1/; (A.7)

so that we can rewrite (A.5) as

@P.n; t/

@tD .L � 1/Œd f.n/P.n; t/� C .L�1 � 1/Œb.n/P.n; t/�: (A.8)

Applying Taylor expansions to a.n C 1/ and a.n � 1/ at n we find that the operators.L � 1/ and .L�1 � 1/ can be written as

.L � 1/Œa.n/� D a.n C 1/ � a.n/ D�a.n/ C a0.n/ C a00.n/

2C � � �

�� a.n/

D1X

kD1

1

kŠ

dka.n/

dnk(A.9)






and

.L�1 � 1/Œa.n/� D a.n � 1/ � a.n/ D�a.n/ � a0.n/ C a00.n/

2C � � �

�� a.n/

D1X

kD1

.�1/k

kŠ

dka.n/

dnk: (A.10)

Using the ansatz (3.40), we will now rewrite (A.8) in terms of �.t/ WD �f.t/ and�.t/ D � f.t/. Observe first that, since �.t/ is assumed to be deterministic, we canwrite

P.n; t/ D Q.�; t/ D Q.�.t/; t/; (A.11)

where Q.�; t/ is the distribution of the stochastic process �.t/. This leads to

@P.n; t/

@tD @Q.�; t/

@tC @Q.�; t/

@�

d�

dt

D @Q.�; t/

@t�

pN

@Q.�; t/

@�

d�

dt; (A.12)

where we differentiated the relation

n D N�.t/ Cp

N �

with respect to t at constant n to obtain

d�

dtD �

pN

d�

dt:

Next observe that the transition probabilities can be expressed as

d.n/ D d.�; t/ D �.N�.t/ Cp

N �/; (A.13)

b.n/ D b.�; t/ D �.N � N�.t/ �p

N �/; (A.14)

where � WD �f and � WD �f . Finally, since a.n/ D a.�; t/ D a.N�.t/ Cp

N �/ wehave that

da.n/

dnD 1p

N

da.�/

d�; (A.15)

so that (A.9) and (A.10) become

.L � 1/Œa.�/� D1X

kD1

1

kŠN k=2

dka.�/

d�k; (A.16)

.L�1 � 1/Œa.�/� D1X

kD1

.�1/k

kŠN k=2

dka.�/

d�k: (A.17)






Inserting (A.12) in the left-hand side of (A.8) and (A.13)–(A.17) in the right-handside, we obtain

@Q

@t�

pN

@Q

@�

d�

dtD .L � 1/Œd.�; t/Q.�; t/� C .L�1 � 1/Œb.�; t/Q.�; t/�

D .L � 1/Œ�.N�.t/ Cp

N �/Q.�; t/�

C .L�1 � 1/Œ�.N � N�.t/ �p

N �/Q.�; t/�

D� 1X

kD1

1

kŠN k=2

dk

d�k

�Œ�.N�.t/ C

pN �/Q.�; t/�

C� 1X

kD1

.�1/k

kŠN k=2

dk

d�k

�Œ�.N � N�.t/ �

pN �/Q.�; t/�:

(A.18)

Collecting terms of orderp

N in the equation above leads to11

d�

dtD � � .� C �/�; (A.19)

whose solution is readily found to be12

�.t/ D �

� C �C e�.�C�/t

��.0/ � �

� C �

�: (A.20)

11 At this point in the derivation, the authors of Di Guilmi et al (2010) inexplicably change thetransition rate (A.13)–(A.14) to the form given in their equation (13A.19), which coincides with thetransition rates for a different model described on p. 23 of Aoki (2002). The analogs of (A.19) and(A.21) thus obtained (Di Guilmi et al 2010) coincide with the corresponding equations on p. 37 ofAoki (2002), but they are not related to the model described in Di Guilmi et al (2010) up to thispoint.12 Up to here, the derivation also works for time-dependent transition rates �.t/ and �.t/. However,the solution (A.20) and the corresponding asymptotic value �1 D �=.�C�/ only hold for constants� and �. The same is true for (13.30) in Di Guilmi et al (2010), which only holds as a solution totheir (13.27) when � and � (their analog of �) are constant, so it is unclear how the authors obtainsteady-state results that depend on �1 (such as the output dynamics in their (13.32)) when thetransition rates are time dependent, as implied by their equations (13.16) and (13.17). This is evenmore problematic for state-dependent transition rates, as suggested in equations (13.33)–(13.34) inDi Guilmi et al (2010), since in this case � and � would be functions of � in (A.18) and would notlead to (A.19) for �.






Similarly, collecting terms of order 1 in (A.18) leads to

@Q

@tD .� C �/

@.�Q/

@�C �� C �.1 � �/

2

@2Q

@�2: (A.21)

We therefore see that � admits an asymptotically stationary distribution

Q1.�/ WD limt!1

Q.�; t/ (A.22)

satisfying@2Q

@�2D � 2.� C �/

��1 C �.1 � �1/

@.�Q/

@�; (A.23)

where

�1 D limt!1

�.t/ D �

� C �: (A.24)

Integrating (A.23), we find that13

Q1.�/ D 1

�p

2�e��2=2�2

; (A.25)

where

�2 D ��

.� C �/2: (A.26)


The authors report no conflicts of interest. The authors alone are responsible for thecontent and writing of the paper.

ACKNOWLEDGEMENTS

Matheus R. Grasselli was partially supported by a Discovery Grant from the NaturalScience and Engineering Research Council of Canada (NSERC) and by the Institutefor New Economic Thinking (INET) through Grant INO13-00011. The hospitalityof the Center for Financial Mathematics and Actuarial Research, University of SantaBarbara, where this work was completed is also gratefully acknowledged. The authorsthank Corrado Di Guilmi, Marco Pangallo and John Muellbauer for comments anddiscussions, as well as the participants of the Econophysics Colloquium (São Paulo,July 2016), the Research in Options Conference (Rio de Janeiro, December 2016),

13 The same remark about transition rates applies here: (A.21) holds for time-dependent rates �.t/

and �.t/ (but not for state-dependent ones), whereas the stationary solution (A.23) only holds forconstants � and �.






the Bachelier Colloquium 2017 (Metabief, January 2017), the INET Researcher Sem-inar (Oxford, March 2017) and the University of Southern California MathematicalFinance Colloquium (Los Angeles, April 2017), where portions of this work werepresented.

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