General Equilibrium Under Convex Portfolio Constraints and Heterogeneous Risk Preferences

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1 General Equlbrum Under Convex Portfolo Constrants and Heterogeneous Rsk Preferences Tyler Abbot arxv: v3 [q-fn.gn] 18 Jun 2018 June 19, 2018 Abstract Ths paper characterzes the equlbrum n a contnuous tme fnancal market populated by heterogeneous agents who dffer n ther rate of relatve rsk averson and face convex portfolo constrants. The model s studed n an applcaton to margn constrants and found to match real world observatons about fnancal varables and leverage cycles. It s shown how margn constrants ncrease the market prce of rsk and decrease the nterest rate by forcng more rsk averse agents to hold more rsky assets, producng a hgher equty rsk premum. In addton, heterogenety and margn constrants are shown to produce both pro- and counter-cyclcal leverage cycles. Beyond two types, t s shown how constrants can cascade and how leverage can exhbt hghly non-lnear dynamcs. Fnally, emprcal results are gven, documentng a novel stylzed fact whch s predcted by the model, namely that the leverage cycle s both pro- and counter-cyclcal. Keywords: Asset Prcng, Heterogeneous Agents, General Equlbrum, Fnancal Economcs. Scences Po, Department of Economcs, 28 rue des Sants Pères, Pars, 75007, France E-mal address: tyler.abbot@scencespo.fr I would lke to thank my advsors Ncolas Cœurdacer and Stéphane Gubaud for ther support durng ths research. I would also lke to thank Georgy Chabakaur for the nsght that motvated the foundaton of ths paper, as well as Semyon Malamud, Julen Hugonner, Ronne Srcar, Gordon Ztkovc, Jean-Franços Chassagneux, Thomas Pumr, Thomas Bourany, Ncolo Dalvt, Rccardo Zago, and Edoardo Gscato for helpful dscussons. Fnally, I should thank the partcpants at the EPFL Brown Bag Semnar, the Scneces Po Lunch Semnar, Prnceton Informal Doctoral Semnar, 2017 RES Meetng, Pars 6/7 MathFProNum semnar, and the SIAM MMF Conference for ther questons and comments. A porton of ths work was funded by an Allance Doctoral Moblty Grant and by a Prnceton-Scences Po PhD Exchange Grant. 1

2 Introducton Market ncompleteness and ndvdual heterogenety are two mportant characterstcs of fnancal markets. Many markets exhbt ncompleteness, n the sense that one cannot freely choose ther portfolo choces ether because of constrants mposed by lenders, because of regulatory constrants, or smply because of a true ncompleteness n the market. At the same tme, n order to generate trade among ndvduals they must dffer n some form. If all agents were dentcal then market prces would make them ndfferent to ther portfolo holdngs and there would never be any trade. Ths paper seeks to combne these two facts about fnancal markets by combnng portfolo constrants and preference heterogenety, wth a partcular applcaton to margn constrants. Margn constrants ncrease the market prce of rsk and decrease the nterest rate, contrbutng to a hgher equty rsk premum. The nterest rate s low because the constrant lmts the supply of rsk free bonds to the market. Ths lmt n supply pushes up the bond prce and down the nterest rate. On the other hand, the market prce of rsk s hgh because constraned agents are unable to leverage up to take advantage of hgh returns. On the opposte sde of ths constrant are rsk averse agents who would lke to sell ther rsky assets to reduce the volatlty of ther consumpton. They are unable to do so, gven the counter-party s the constraned agent. Thus margn constrants create an mplct lqudty constrant whch allows the market prce of rsk to reman hgh n order to compensate rsk averse agents for havng a rsker consumpton stream. Asset prces are hgher or lower than n an unconstraned equlbrum dependng on whether the ncome effect or the substtuton effect domnates. When agents are constraned, other agents are forced to hold more rsky assets. These unconstraned agents hold both rsky and rsk-free assets, mplyng that an ncrease n the market prce of rsk and a decrease n the rsk-free rate represent an ambguous change n the nvestment opportunty set. However, the effect tends to ncrease the equty rsk premum. Ths has the effect of ncreasng the dscount rate and smultaneously makes agents wealther today and makes consumpton tomorrow less expensve. The frst effect an ncome effect) causes agents to desre to consume more today. The second effect a substtuton effect) causes agents to desre to consume less today and more tomorrow. Whch effect domnates depends on the EIS of unconstraned agents. If EIS s less than one then the substtuton effect domnates: ndvduals consume less today, pushng up ther wealth and thus ncreasng asset prces. If EIS s greater than one then the ncome effect domnates: ndvduals consume more today, pushng down ther wealth and thus reducng asset prces. In ths way we can see ether an ncrease or a decrease n asset prces when some porton of agents are constraned, dependng on whether the EIS s greater 1

3 or less than one. Margn constrants and preference heterogenety generate both pro- and counter-cyclcal leverage cycles. Less rsk averse agents domnate the economy and the prce of rsky assets s hgh when aggregate producton s hgh. Hgh asset prces ncrease ndvdual wealth and reduce leverage. On the contrary, rsk averse agents domnate when aggregate producton s low, reducng asset prces. Low asset prces cause ndvdual wealth to be low and ndvdual leverage to be hgh. Wth the ntroducton of a margn constrant less rsk averse agents eventually run nto a borrowng lmt. Not only s borrowng reduced, but, as dscussed before, asset prces can be hgher under constrant. In turn, total leverage falls. In ths way heterogeneous preferences and margn constrants produce both pro- and counter-cyclcal leverage cycles. Fnancal leverage has become an mportant polcy varable snce the crss of In partcular leverage allows nvestors to ncrease the volatlty of balance sheet equty, producng the possblty of greater returns. At the same tme leveraged nvestors are exposed to larger down-sde rsk. In the face of negatve shocks, constraned nvestors must sell assets to reduce ther leverage. Ths s known as the leverage cycle. The assocated credt contracton produces large volatlty n asset prces and has been the target of regulaton n the post-crss era e.g. the Basel III captal requrement rules). However, leverage cyclcalty remans a topc of debate. Leverage cyclcalty s both pro- and counter- cyclcal n the model presented here, dependng on the aggregate state of the economy and the margnal agent. In secton 4 I document n the data that cycles are both pro- and counter-cyclcal dependng on the level of aggregate asset prcng varables whch can be nterpreted as proxes for margnal preferences. Ths fact could reconcle some of the emprcal debates about the cyclcalty of leverage and re-enforces the study of preference heterogenety as a drver of fnancal trade. Convex portfolo constrants arse qute naturally n fnance. A convex constrant smply states that the portfolo weghts must le n a convex set contanng zero see Stgltz and Wess 1981) for an example of mcro-foundatons to credt constrants). In macroeconomcs there are countless examples of partcular models wth market ncompleteness whch can be descrbed n ths settng of convex constrants, such as Ayagar 1994); Kyotak and Moore 1997); Krusell and Smth 1998); Bernanke et al. 1999) and many others. A margn constrant essentally states that an agent cannot borrow nfntely aganst ther equty. Ths type of constrant s seen n consumer fnance when borrowng money to purchase a home: one must almost always put up a down payment. In fnancal markets margn constrants arse n repo markets and other lendng vehcles see Hardouvels and Perstan 1992); Hardouvels and Theodossou 2002); Adran and Shn 2010a) for emprcal studes of margns). In fact real world experence motvated the theoretcal study of leverage cycles 2

4 ntated by Geanakoplos 1996). In addton lmts to arbtrage and fnancal bubbles have been studed under margn constrants n the context of lqudty see e.g. Brunnermeer and Pedersen 2009)). Many of these phenomena arse n the model presented n ths paper, but the predctons for leverage cycles are emphaszed because of ther novelty. In theoretcal models leverage cyclcalty depends greatly on the underlyng assumptons producng trade. In hs foundatonal work on the topc, Geanakoplos 1996) shows how the combnaton of belef heterogenety and margn constrants produce a pro-cyclcal leverage cycle. However, ths fndng s n opposton to the contemporary paper by Kyotak and Moore 1997), where partcpaton constrants force agents to nvest through ntermedares, whose credt constrants produce a counter-cyclcal leverage cycle. More recently He and Krshnamurthy 2013) and Brunnermeer and Sannkov 2014) also produce counter-cyclcal leverage cycles by ncludng ntermedares through whom constraned agents can proft from rsky assets. In fact, He and Krshnamurthy 2013) even ponts out the debate n the appled lterature and the fact that, [Ther] model does not capture the other aspects of ths process,... that some parts of the fnancal sector reduce asset holdngs and deleverage. These models mply that the mechansm producng trade determnes leverage s cyclcalty. The emprcal lterature has noted ths ambguty over the cyclcalty of leverage n dfferent. Korajczyk and Levy 2003) study the captal structure of frms and fnd that leverage s counter-cyclcal for unconstraned frms and pro-cyclcal for constraned frms. However, Hallng et al. 2016) contradct ths by showng that target leverage s counter-cyclcal once you account for varaton n explanatory varables, pontng out that the effect n Korajczyk and Levy 2003) s only the drect effect. In the cross secton of the economy Adran and Shn 2010b) fnd that leverage s counter-cyclcal for households, ambguous for non-fnancal frms, and pro-cyclcal for broker dealers. However, the authors study the relatonshp between leverage and changes n balance sheet assets. Ths comparson produces a mechancal correlaton whch somewhat dsappears when assets are replaced by GDP growth as a proxy for the busness cycle see Fgure 13). Ang et al. 2011) pont out that when accountng for prces broker dealer leverage s counter-cyclcal, but that hedge fund leverage s pro-cyclcal. These contrary studes can be reconcled when controllng for fnancal varables such as the prce/dvdend rato or the nterest rate. In fact, for several sectors studed see secton 4) the leverage cycle s both pro- and counter-cyclcal. Ths ambguty s predcted by the model of preference heterogenety and margn constrants presented here. Many authors have crtczed the assumpton of a representatve, constant relatve rsk averson agent snce Mehra and Prescott 1985) posted the equty rsk premum puzzle. The defnton of new utlty functons was the frst major response to ths puzzle, n partcular Epsten-Zn preferences Epsten and Zn 1989); Wel 1989)) and habt formaton Campbell 3

5 and Cochrane 1999)) have been used to explan the equty premum puzzle. However, several papers have studed preferences across ndvduals and found them to be heterogeneous and constant n tme Brunnermeer and Nagel 2008); Chappor and Paella 2011); Chappor et al. 2012)), contradctng both of these new branches of the theoretcal lterature. In addton, Epsten et al. 2014) ponted out that the assumptons necessary to match the rsk premum usng Epsten-Zn preferences produce unrealstc preference for early resoluton of uncertanty. Beyond these crtcsms, one needs heterogenety n order to generate trade at all n any market model. In a representatve agent settng one looks for the prces whch make the agent ndfferent to not tradng. Rsk preference heterogenety has succeeded n partally respondng to these ssues. Heterogenety n rsk preferences has been used to generate trade n fnancal models snce the foundatonal paper of Dumas 1989). Snce then many authors have studed the problem from dfferent angles, assumng dfferent levels of market completeness, utlty functons, partcpaton constrants, nformaton structures, etc., but almost always under the assumpton of only two preference types Basak and Cuoco 1998); Coen-Pran 2004); Guvenen 2006); Kogan et al. 2007); Guvenen 2009); Cozz 2011); Garleanu and Pedersen 2011); Hugonner 2012); Rytchkov 2014); Longstaff and Wang 2012); Preto 2010); Chrstensen et al. 2012); Bhamra and Uppal 2014); Chabakaur 2013, 2015); Gârleanu and Panageas 2015); Santos and Verones 2010)). Cvtanć et al. 2011) studes the problem of N agents wth several dmensons of heterogenety and focuses on the domnant agents, characterzng portfolos va the Mallavan calculus. Abbot 2017) studes a settng wth N heterogeneous CRRA agents n a complete fnancal market usng a value functon approach and shows how changes n the number of types can produce substantally dfferent quanttatve results and how the varance n preferences provdes an addtonal degree of freedom for explanng the equty rsk premum puzzle. However, that work produces large amounts of aggregate leverage and hgh ndvdual margns. Ths observaton ponts towards the need to ntroduce some degree of constrant or ncompleteness to better match the real world. To that end, ths paper studes the same type of economy wth N heterogeneous CRRA agents under convex portfolo constrants wth an applcaton to margn constrants. A fundamental paper by Cvtanć and Karatzas 1992) studed the general case of convex portfolo constrants n partal equlbrum. The authors developed an ngenous way to embed the agent n a seres of fcttous economes, parameterzed va a sort of Kuhn-Tucker condton, and then to select the approprate market to make the agent just ndfferent. However, ther approach was to use convex dualty to characterze the soluton, whch reles on a strct assumpton that the relatve rsk averson be bounded above by one. Ths lmtaton led others to look to solve the prmal problem drectly, such as He and Pages 4

6 1993); Cuoco and He 1994); Cuoco 1997); Karatzas et al. 2003). These works use dense and complex mathematcal technques whch may or may not provde tractable solutons for calculaton. The present paper takes a more drect approach to solve the prmal problem by notcng that homogeneous preferences are assocated to a value functon whch factors nto a functon of wealth and a functon of the aggregate state, under the approprate ansatz. Usng ths ansatz, the Hamlton-Jacob-Bellman equaton becomes a PDE over consumpton weghts. 1. A Model of Preference Heterogenety 1.1. Fnancal Markets Consder a contnuous tme, nfnte horzon Lucas 1978) economy wth one consumpton good. Ths consumpton good, denoted D t, s produced by a tree whose dvdend follows a geometrc Brownan moton GBM): dd t D t = µ D dt + σ D dw t where W t s a standard Brownan moton and µ D, σ D ) are constants. Agents can trade n a locally) rsk-free and a rsky securty, whose prces are denoted St 0 and S t respectvely. These prces are assumed to follow an exponental and an Itô process, respectvely: ds t S t = µ t dt + σ t dw t 1) ds 0 t S 0 t = r t dt 2) where µ t, σ t, r t ) are determned n equlbrum. Indvduals are ntally endowed wth a share n the per-capta tree, α 0, and a poston n the rsk-free asset, β Preferences and Wealth The economy s populated by an arbtrary number N of atomstc agents ndexed by {1,..., N}. Agents have constant relatve rsk averson CRRA) preferences and dffer n ther rate of relatve rsk averson, γ, such that ther nstantaneous utlty s gven by u c) = c1 γ 1 γ 5

7 Denote by X t an ndvdual s wealth at tme t and note that ntal wealth s gven by X 0 = α 0 S 0 + β 0 St 0. Denote by π t the share of an ndvdual s wealth nvested n the rsky stock, whch mples 1 π t s the share nvested n the bond. Assumng that tradng strateges are self fnancng, an ndvdual s wealth evolves as dx t = [X t r t + π t µ t + D )) ] t r t c t dt + X t π t σ t dw t S t 1.3. Portfolo Constrants and Indvdual Optmzaton Indvdual nvestors solve a utlty maxmzaton problem subject to ther self-fnancng budget constrant and a portfolo constrant: max {c t,π t } t=0 E s.t. dx t = 0 π t Π e ρt c1 γ t dt 1 γ [X t r t + π t µ t + D t r t S t )) ] c t dt + X t π t σ t dw t where Π R represents a closed, convex regon of the portfolo space whch contans {0}. For example Π = R s the unconstraned case, Π = R + s a short sale constrant, Π = {π : π m m 0} s a margn constrant. Ths set s allowed to dffer across agents, as mpled by the subscrpt. Ths paper focuses on an applcaton to margn constrants, but the approach s applcable to any constrant whch can be wrtten as a functon of the aggregate state Equlbrum Investors are consdered to be atomstc and thus I consder a Radner 1972) type equlbrum. Defnton 1. An equlbrum n ths economy s defned by a set of processes {r t, S t, {c t, X t, π t } N =1} t, gven preferences and ntal endowments, such that {c t, X t, π t } solve the agents ndvdual optmzaton problems and the followng set of market clearng condtons s satsfed: c t = D t, 1 π t )X t = 0, π t X t = S t 3) I study Markovan equlbra such that equlbrum quanttes can be wrtten as functons of some state vector. That s for some equlbrum process Y t, I look for functons f ) such 6

8 that Y t = fz t ) for some process z t. I wll look for a partcular equlbrum n the vector of consumpton weghts defned by ω = [ω 1t,..., ω N 1)t ] T = [c 1t /D t,..., c N 1)t /D t ] T Ths s n the sprt of Chabakaur 2013, 2015), where gven two agents we can take the consumpton weght of a sngle agent as the state varable. I ve only ncluded N 1 consumpton weghts because the last s determned by market clearng. However, for some equatons the full vector of weghts s useful, so I wll defne Ω = [ω 1t,..., ω Nt ] T = [c 1t /D t,..., c Nt /D t ] T The followng secton wll descrbe how to characterze equlbrum processes n terms of these quanttes. 2. Equlbrum Characterzaton To solve ths problem I begn wth the approach of Cvtanć and Karatzas 1992). Ths method uses a fcttous, unconstraned economy and a shadow cost of constrant, or Lagrange multpler, to fnd the correct prcng process. Unlke n ther work I do not use a dualty approach, but show how the prmal problem admts a Markov representatve. The reason ths works s because when preferences are homothetc they can be represented by a utlty functon whch s homogeneous of some degree. In ths case the value functon factors and the resultng ODE s no longer a functon of ndvdual wealth. Ths approach wll lkewse work for any homogeneous utlty functon, ncludng Epsten-Zn 1. descrbed n the followng subsectons. Ths process wll be 1 In partcular, frst order condtons from a dynamc program gve consumpton as c = u 1 X JX, Y )), where Y s any arbtrary, aggregate state vector and X an ndvdual s wealth. We would lke to fnd c = X/V Y ). Equate these and rearrange to fnd X JX, Y ) = u X/V Y )). When the utlty functon s homogeneous of degree k + 1, u ) s homogeneous of degree k. Thus X JX, Y ) = u 1)V Y ) k X k. By ntegratng wth respect to X one fnds a proposal for the value functon such that consumpton s a lnear functon of wealth. 7

9 Π δ ν) N Unconstraned R 0 0 Margn Constrant {π : π m ; m 0} νm R Short-Sale Constrant {π : π s ; s 0} νs R + Table 1: Examples of constrant sets, support functons, and effectve doman of the adjustment ν Optmalty n Fcttous Unconstraned Economy In order to fnd the constraned equlbrum, we defne new processes for ndvdual prces, whch are adjusted by a process ν t, consdered the shadow cost of constrant: dst 0 St 0 ds t S t = r t + δ ν t ))dt = µ t + ν t + δ ν t ))dt + σ t dw t The functon δ ) s the support functon of Π, whch s defned as δ ν) = sup π Π νπ) In addton, ths gves rse to the effectve doman of ν t defned by N = {ν R : δ ν) < } for examples see Table 1). Fnally, we have a complmentary slackness condton whch states ν t π t + δ ν t ) = 0. Each agent solves ther optmzaton problem n the face of ther ndvdual, fcttous fnancal market. Defne the stochastc dscount factor SDF) of an ndvdual agent as an Itô process whch evolves as a functon of the ndvdual s adjustment: dh t H t = r t + δ ν t ))dt θ t + ν ) t dw t 4) σ t By a straght-forward applcaton of the martngale approach Karatzas et al. 1987)) n ths fcttous economy one fnds ndvdual consumpton as a functon of ndvdual SDF s: c t = Λ e ρt H t ) 1 γ 5) for all, where Λ s the Lagrange multpler assocated to the statc budget constrant. In the case where Π = R, the SDF s concde and the ratos of margnal utltes are constant. 8

10 However, when agents are constraned n ther portfolo choce ths s not the case and we have c γ t c γ j jt = Λ H t Λ j H jt These ratos of SDF s, whch are proportonal to ratos of margnal utltes, are very famlar n the theory of ncomplete market equlbra. In Cuoco et al. 2001), a representatve agent wth state dependent preferences s studed, where the preferences are a weghted average of ndvdual preferences. The stochastc weghts are exactly equal to the rato of margnal utltes. Ths s also seen n Basak and Cuoco 1998) and Hugonner 2012) General Equlbrum Characterzaton Equlbrum s characterzed by frst assumng the exstence of a Markovan equlbrum, dervng a system of ODE s for wealth-consumptons ratos, then recoverng the adjustments ν t usng the complmentary slackness condtons. Gven ths t s possble to prove optmalty of the value functons 2. Frst, consder the nterest rate and market prce of rsk: Proposton 1. The nterest rate and market prce of rsk can be shown to be functons of weghted averages of ndvduals consumpton weghts, preference parameters, and adjustments such that θ t = 1 r t = 1 ω t γ ω t γ σ D 1 σ t µ D + ρ 1 2 ω t ν t γ ω t γ 1 + γ γ 2 ) ω t 6) δ ν t ) 7) γ θ t + ν ) 2 t ω t) 8) σ t The nterest rate and market prce of rsk take a typcal form, but are augmented by the adjustment to ndvduals margnal utltes. Frst notce that the market prce of rsk Eq. 6)) s determned by the fundamental volatlty σ D dvded by the weghted average of elastcty of ntertemporal substtuton EIS), exactly as n complete markets Abbot 2017)). In addton the constrant wll ether ncrease or reduce the market prce of rsk, dependng on the doman of ν t. In the case of margn constrants ν t 0, so the market prce of rsk wll be weakly hgher under constrant. Ths s drven by an mplctt lqudty constrant. 2 Ths remans a clam at ths pont. The proof s ongong. 9

11 Constraned agents are unable to take advantage of hgh returns. In addton, the effect of volatlty mples that n tmes when stock prce volatlty s low, greater constrant mples greater returns. Ths correlaton s agan drven by the fact that agents cannot borrow to take advantage of the returns, producng the same type of lqudty effect descrbed n the lmts-to-arbtrage lterature e.g. Brunnermeer and Pedersen 2009) or Hugonner 2012)). Rsk neutral agents would arbtrage away the hgh returns, but cannot because of ther margn constrant. The nterest rate smlarly exhbts a famlar shape. We see a rate of tme preference term, an ntertemporal smoothng term, and a prudence or rsk preference term: r t = ρ }{{} Rate of Tme Preference ω t + µ D γ δ ν t ) ω t 1 γ }{{ 2 } Intertemporal Smoothng 1+γ γ 2 θ t + ν t σ t ) 2 ωt ω t γ } {{ } Prudence/Rsk Preferences Both the ntertemporal smoothng and prudence terms are augmented by the constrant. Under a homogeneous margn constrant, δ ν t ) = mν t, but recall that ν t 0, whch together mply that the constrant reduces nterest rates through the ntertemporal smoothng term. Constraned agents are unable to supply bonds to the market n order to transfer consumpton and wealth from the future to today. A lower supply of bonds pushes up the prce and down the nterest rate. At the same tme constrant affects the nterest rate through the prudence motve by changng the demand for precautonary savngs. Indvduals demand more precautonary savngs when ther SDF s more volatle Kmball 1990)). When agents are constraned, ther SDF s less volatle as they are unable to ncrease ther exposure to fundamental rsk. Ceterus parbus, ths reduces the demand for precautonary savngs and ncreases the nterest rate, counteractng the ntertemporal motve. Together these forces produce an equty rsk premum whch depends on the shape of heterogenety, the degree of constrant, and the state varable, all drven by the ndvdual consumpton weghts whch determne the margnal agents. How consumpton weghts evolve over tme s mportant not only from an economc perspectve, but also n order to derve the soluton of the model. We can study the dynamcs of consumpton weghts by applyng Itô s lemma and matchng coeffcents to fnd ther drft and dffuson: 10

12 Proposton 2. Consumpton weghts follow an Itô process whose dynamcs are gven by: dω t ω t where = µ ωt dt + σ ωt dw t µ ωt = 1 γ r t + δ ν t ) ρ γ γ θ t + ν ) 2 t σ D θ t + ν ) ) t σ t σ t + σd 2 µ D 9) σ ωt = 1 θ t + ν ) t σ D 10) γ σ t Ths mples that the state varable ω follows an Itô process such that dω = µ ω dt + σ ω dw t 11) where µ ω = [µ ω1t ω 1t,..., µ ωn 1)t ] T and σ ω = [σ ω1t ω 1t,..., σ ωn 1)t ] T These equatons are very smlar to those one fnds n the complete markets case Abbot 2017)), but augmented by the constrant. In partcular, consder the volatlty of consumpton weghts gven n Eq. 10). An agent s consumpton volatlty s exactly zero when ther preference parameter satsfes γ = 1 Ξ t 1 + ν t where ξ t = ξ t ξ t σ D σ D σ t ω t γ, Ξ t = ω t ν t γ We can thnk of ths as the margnal preference level n the market for consumpton. However, t s possble that ths preference level s not unque. Consder the case where some agents face a margn constrant, but others do not. Amongst the unconstraned agents, the margnal preference level corresponds to the frst two terms, whle among the constraned agents all of the terms matter. Gven ν t 0 under margn constrants, there could very well exst both a constraned and an unconstraned agent who have zero consumpton volatlty. Ths s drven by the constraned agents beng unable to leverage up to gan more exposure to aggregate rsk. Snce θ t and r t are functons of {ω t } N =1, t remans to show that {ν t } N =1 are as well. Frst, one can derve a system of PDE s for ndvdual wealth/consumpton ratos, from whch one can determne the adjustments. Proposton 3. Gven Propostons 1 and 2 and assumng adjustments and volatlty are functons of ω such that ν ω) = ν t and σω) = σ t, t s possble to defne the nterest rate 11

13 and market prce of rsk as functons of ω such that r t = rω) and θ t = θω). Assumng there exsts a Markovan equlbrum n ω, the ndvduals wealth-consumpton ratos, V ω) = X t /c t, satsfy PDE s gven for each by [ 1 1 γ )rω) + δ ν ω))) ρ + 1 γ θω) + ν ) ] 2 ω) V ω)+ γ 2γ σω) [ 1 γ θω) + ν ) ] ω) σω T + µ T ω V ω) + 1 σω) 2 σt ωhv ω)σ ω + 1 = 0 γ where and H ω represent the gradent and hessan operators, and where µ ω and σ ω are gven n Proposton 2. 12) Boundary condtons when a sngle agent domnates are gven by the autarkcal case, where lm V jω) = ω 1 ρ 1 γ j ) γ j θω)+ν ω)/σω)) 2 2γ j ) + rω) + δ j ν j ω)) 13) Boundary condtons when an agent s weght goes to zero are gven by the soluton to an N 1 agent problem. The boundary condtons when a sngle agent domnates represent the vertces of the state space. On the other hand, when an agent s weght goes to zero, the economy soluton s equvalent to a two agent economy, wth the zero agent s wealth/consumpton rato stll satsfyng Eq. 12), but ther choces havng no effect on aggregate varables. These partal dfferental equatons represent the shape of ndvduals wealth/consumpton ratos over the state space. Unlke n complete markets, however, the system s hghly non-lnear, snce the coeffcents depend n a complcated way on the soluton tself. Next, consder the portfolos of ndvduals, gven n Proposton 4. Proposton 4. Assumng adjustments and volatlty can be wrtten as functons of ω such that ν ω) = ν t and σω) = σ t, t can be shown that portfolos are functons of ω such that π ω) = π t, where π ω) = 1 θω) + ν ) ω) γ σω) σω) + γ σ ω ω) T V ω) V ω) where σ ω ω) = [σ ω ω)ω ] T s the vector of dffusons of ω. One can see rght away that portfolos take the typcal ICAPM form Merton 1971)). There s frst a myopc term, represented by the market prce of rsk scaled down by rsk averson and 14) 12

14 volatlty, whch gves the nstantaneous portfolo demand of an ndvdual gven the market prce of rsk. Next s a hedgng term, determned by the co-movement of an ndvdual s wealth wth the aggregate state. Fnally, there s a constrant term, whch compensates the ndvdual s portfolo such that they are wthn the constrant set. On an aggregate level, we can derve asset prcng varables from an applcaton of Itô s lemma and from market clearng for wealth. Proposton 5. Assumng adjustments can be wrtten as functons of ω such that ν ω) = ν t, t can be shown that volatlty and the prce dvdend rato are functons of ω such that σω) = σ t and Sω) = S t /D t, where σω) = σ D + σ ωω) T V ω) + J V ω) T Ω ) 15) Sω) where J V ω) represents the Jacoban matrx and where Sω) = ω V ω) 16) represents the prce dvdend rato S t /D t. Volatlty n Eq. 15) s drven by the fundamental volatlty, the shape of wealth consumpton ratos, and the volatlty of consumpton weghts. When agents have hgh volatlty n consumpton weghts, the volatlty of asset prces wll be hgher. At the same tme, ndvduals wealth wll be less volatle under constrant. Ths wll produce a reducton n volatlty. We wll see these two forces n the numercal smulatons n secton 3. We need to derve an expresson for {ν t } N =1 n order to close the model. The functonal form depends on the type of constrant. To that end, I wll focus from here only on margn constrants. The followng proposton gves the functonal form for the adjustments under homogeneous margn constrants when π t m for all, where m 0, whch mples an effectve doman of N = {ν : ν 0} and a support functon of δ ν) = mν. Proposton 6. Under margn constrants, adjustments can be wrtten as functons of ω such that ν ω) = ν t, where ν ω) = mn {0; mγ σω) 2 1 θω) 1 + σ ))} ωω) T V ω) mσω) γ V ω) Fnally, we need a verfcaton argument for optmalty of the value functons. In partcular, we would lke to be sure that soluton to the PDE s n Proposton 3 are ndeed the wealth/consumpton ratos assocated to the ndvduals optmal choces. If we are wllng 13 17)

15 to make the assumpton that the wealth/consumpton ratos are twce contnuously dfferentable, then we can easly show usng Itô s lemma that the value functons are ndeed optmal ths proceeds as n Chabakaur 2015)). However, t would be preferable to relax ths assumpton. To do so we can make use of a powerful new result n Confortola et al. 2017), namely that under certan condtons the value functon mpled by Proposton 3 s ndeed optmal 3. Clam 1. Assumng that ndvdual wealth/consumpton ratos V ω) are C 1 wth bounded frst dervatve, then there exsts a unque soluton to Eq. 12) n the vscosty sense) and ths soluton corresponds to the value functons n Eq. 22). Furthermore ths represents a Markovan equlbrum satsfed by Propostons 1 to 6. Ths clam reles only on a sngle degree of dfferentablty Numercal Soluton Ths secton presents numercal results for several assumptons about the dstrbuton of preferences. Frst, the case of two types s evaluated and the cyclcalty of the leverage cycle s emphaszed. The leverage cycle s pro- or counter-cyclcal dependng on the margnal agent. Second, results are presented for three agents. Two key features whch are not observed n the two agent case are the possblty of cascadng constrants and a hghly non-monotonc leverage. When one agent s constraned, other agents tend to hold more leverage. Ths pushes the ntermedate agent closer to ther own constrant. At the same tme, ths ncrease n ndvdual leverage can partally or even fully offset the reducton n total leverage generated by the frst agent s constrant. Over all smulatons I hold fxed µ D, σ D, ρ) = 0.01, 0.032, 0.02), chosen to compare to Chabakaur 2015) Two Types and Leverage Cycles Consder the case of two agents 5 wth relatve rsk averson γ 1, γ 2 ) = 1.1, 5.0) who face a margn constrant such that the share, π t, of ther wealth nvested n the rsky asset s less 3 Ths s left as a clam, as only an outlne of a proof has been completed. 4 I beleve ths condton can be relaxed to smply Lpschtz contnuty. 5 The two agent model represents the boundary of the three agent problem, so s ts soluton s necessary to treat the three agent case. In addton, understandng the shape of functons n ths smple case wll help to fx deas n the more complex case of arbtrary number of types. 14

16 than some constant m. Ths s equvalent to a leverage constrant: π t m α ts t α t S t + b t S 0 t m b ts 0 t α t S t m b ts0 t α t S t m 1 m In partcular, take m = m = 1.2. There wll exst a regon of the state space over whch ths constrant bnds for the less rsk-averse agent. In ths regon, the more rsk averse agent holds a larger share of ther wealth n the rsky asset. In order to acheve these portfolo weghts, the constraned agent holds fewer rsky shares and the unconstraned agent holds more rsky shares. By reducng ther rsky shares, the less rsk-averse agent s constrant actually tghtens, causng them to sell more rsky-shares, makng the effect more than proportonal. Ths corresponds to a substantal declne n leverage and a tghtenng of credt demand, pushng down the nterest rate. In addton, the market prce of rsk s hgh n order to compensate the rsk averse nvestor for holdng a larger share. Whether these two effects combne to make asset prces hgher or lower depends on whether the ncome or wealth effect domnates. Portfolos are represented n Fgure 1a). Movng form rght to left n the state space, agent 1, the least rsk averse agent would prefer to leverage up, but runs nto ther constrant. In order reman below ther constrant they adjust the composton of ther wealth. The portfolo weght s fallng n rsk free borrowng, so the agent reduces ther rsk free borrowng. Ths reducton n the supply of rsk free assets pushes up the prce and down the nterest rate, as s seen n Fgure 2a). But how does the agent fnance a reducton n ther borrowng? They shft ther wealth out of rsky shares and nto rsk-free savngs. When the agent s a borrower the portfolo weght s decreasng n rsky shares, so ths actually can only partally allevate ther stuaton. The constraned agent must further reduce ther borrowng. As seen n Fgures 1c) and 1d), there s a substantal fall n borrowng and leverage as ths agent shfts out of rsky assets and nto rsk-free assets. The market prce of rsk must be hgher to compensate the unconstraned agent for holdng more rsky assets. As prevously mentoned, the constraned agent s sellng rsky assets to the unconstraned agent, who s more rsk averse. Ths agent requres hgher returns on the rsky asset and so the market prce of rsk s hgher Fgure 2b)). The combnaton of a lower nterest rate and hgher market prce of rsk produces an ambguous effect on the rsky asset prce. To dscuss asset prces we need to consder how ndvdual preferences translate nto choces about consumpton gven changes n the nvestment opportunty set. Gven an mprovement n the nvestment opportunty set, an agent wll always have a substtuton effect whch reduces consumpton today, as they substtute consumpton from today to tomorrow. 15

17 a) b) c) d) Fg. 1. On the other hand, the agent s rcher today and gets ncome from ther wealth, mplyng an ncome effect. Ths ncome effect pushes up consumpton n all perods. The nteracton of these two forces determne the level of consumpton today, whch n turn determnes the wealth/consumpton rato and asset prce. Gven an mprovement n the nvestment opportunty set, the prce of the rsky asset ncreases or decreases dependng on whether the ncome or substtuton effect domnates. When an agent has RRA of one, or EIS of one, ther ncome and substtuton effects perfectly 16

18 a) b) c) d) Fg. 2. offset. When EIS s less than one the ncome effect domnates and the agent chooses to ncrease consumpton today. Thus, relatve to wealth, ther consumpton s greater, mplyng a lower wealth/consumpton rato. Ths reduces asset prces whch are a weghted average of wealth/consumpton ratos. When EIS s greater than one the substtuton effect domnates. Wealth/consumpton ratos rse and the asset prce ncreases. In the present settng, both agents have low EIS, so we expect that for a gven mprovement/deteroraton n the nvestment opportunty set, asset prces wll be lower/hgher. 17

19 Fgures 2c) and 2d) show that over part of the state space the asset prce s ndeed hgher under constrant. Although the market prce of rsk s hgher, the rsk free rate s lower and the constrant shfts more weght to agents who are net lenders. Thus the effect on the rskfree rate domnates and the nvestment opportunty set deterorates and, snce the ncome effect domnates, the asset prce rses. However, n the lower area of the state space the asset prce s lower than n the absence of constrant. Ths s drven by the fact that the change n the nvestment opportunty s not unambguously negatve. The unconstraned agent holds both rsky and rsk-free assets, and the return on rsk-free assets has fallen. To see how the relatve returns on these two assets changes we can look at the equty rsk premum ERP t = µ t + D t S t r t = θ t σ t The equty rsk premum s the expected captal gans plus dvdend yeld mnus the rsk free rate, whch s smply the market prce of rsk tmes volatlty. In Fgure 3c) we see that the asset prce s lower or hgher under constrant exactly when the equty rsk premum s lower or hgher. Ths s because the unconstraned agent s a net lender, so s essentally short the equty rsk premum. Any mprovement n ths premum translates to a deteroraton n the nvestment opportunty set faced by the unconstraned agent, causng them to reduce consumpton and pushng up ther wealth/consumpton ratos and, n turn, asset prces. In addton to these frst-order moments, the dynamcs of the model are also affected by the constrant. There s a reducton n trade when one agent s constraned. In ths regon, shares are only exchanged n order to mantan the portfolo weght whch holds the constraned agent aganst ther constrant. Ths reducton n exchange dampens volatlty as there s less change n the margnal agent prcng rsky assets. Ths effect can be seen n Fgure 3b). One take-away from ths observaton could be an ntuton for volatlty frowns and smrks observed n optons prcng data. Impled volatlty can have a postve or negatve term premum for dfferent values of the strke, mplyng changes n volatlty over the state space. What ths model predcts s that for markets where partcpants face constrants n ther trade of the underlyng, there wll be a postve term premum or frown), whle for unconstraned assets there wll be a negatve term premum smrk or smle). The effect on leverage s substantal gven both a supply effect and a demand effect. The demand for credt s artfcally lower under constrant when rsk neutral agents cannot leverage up. The supply of credt s also reduced because rsk averse agents shft wealth nto rsky assets. They do so because they see low volatlty and hgh expected returns. Rsk averse agents shft wealth nto rsky shares and the supply of credt contracts. As the economy moves between the constraned and unconstraned regons the cyclcalty of leverage 18

20 a) b) c) Fg. 3. changes. Leverage cycles are both pro- and counter-cyclcal n both complete and ncomplete markets, but the dynamcs of ths cyclcalty s vastly dfferent under the two regmes. In complete markets, the slope of leverage vares smoothly, movng from postve to negatve as one moves through the state space. Only n very bad states does leverage exhbt procyclcalty, as rsk averse agents begn to domnate and the nterest rate becomes too hgh for rsk neutral agents to desre to borrow. Ths nflecton pont becomes a sngularty 19

21 under margn constrants. In Fgure 1d) we can see a knk at the boundary between the constraned and unconstraned regons, mplyng a jump from pro- to counter-cyclcalty. Ths predcton connects to the large lterature on the cyclcalty of leverage Geanakoplos 1996, 2010); Adran and Shn 2010b), as well as many others). These observatons wll be studed emprcally n secton Three Types, Cascadng Constrants, and Non-Monotonc Leverage Cycles Consder next the case of three agents 6. Introducng a thrd agent shows how there can exst a cascade effect. As the least rsk-averse agent s constrant bnds, the other agents begn to leverage up. Ths causes the agent n the mddle of the dstrbuton of preferences to move towards ther constrant. Ths s not evdent wth only two agents, as the most rsk-averse agent wll never ht ther constrant. The ncrease n leverage of the ntermedate agent actually leads to a full recovery of leverage. That s, n the regon where the least rskaverse agent s constraned, the ntermedate agent wll take ther place n the market for borrowng, partally or even fully offsettng the reducton n borrowng caused by constrant. Ths leads to a sort of double-dp n leverage: frst leverage contracts as one agent becomes constraned, then rses as the ntermedate agent takes up the slack, and eventually falls when the ntermedate agent also becomes constraned. All parameters are the same as n secton 3.1, except preferences whch are set to γ 1, γ 2, γ 3 ) = 1.1, 1.5, 3.0) and margn constrants are set to m = 1.2 for all. Graphs are plotted over the state space where ω 1, ω2) {x, y) R + : x + y 1}, the two dmensonal smplex. In the extreme cases where D t 0 or D t, ω 1, ω 2 ) 0, 0) and ω 1, ω 2 ) 1, 0), respectvely. Thus we can thnk of negatve shocks pushng n a southwest drecton and postve shocks pushng n a southeast drecton, wth some devaton n the nteror of the state space 7. Consder frst the nterest rate and market prce of rsk, depcted n Fgures 4 and 5. These two varables determne the nvestment opportunty set, whch makes them key n determnng asset prces. You ll notce frst that the nterest rate s ncreasng under negatve shocks. Ths s smlar to the complete market, where negatve shocks push more weght to the most rsk averse agent who s very patent and n turn requres a hgher nterest rate. However we can see that there s a regon where the nterest rate s lower n the constraned equlbrum, evdenced by the negatve values n Fgure 4b). In ths regon at least one agent 6 For a descrpton of the numercal soluton to ths problem see?? B. 7 Quver plot of shock drectons to be added. 20

22 s constraned. There s a contracton n the demand for credt, pushng up the prce of bonds and down the nterest rate. The market prce of rsk follows smlar dynamcs Fgure 5) for smlar reasons. However, the market prce of rsk s hgher under constrant, as seen n Fgure 5b). Ths s drven by rsk-averse agents requrng hgher returns to hold a greater share of ther wealth n rsky assets. These effects combned have an ambguous effect on asset prces, a pror, but tend to ncrease asset prces for the gven parameterzaton. a) b) Fg. 4. Rsk free rate n levels Fgure 4a)) and n devatons from complete markets??). Asset prces are hgher under constrant, as can be seen n Fgure 6. Ths effect s smlar to that dscussed n secton 3.1. Because agents have EIS less than one, the ncome effect 21

23 domnates and wealth consumpton ratos ncrease, pushng up asset prces. Interestngly, the devaton n asset prces s very steep near the boundary, drven by a rapd deteroraton n the equty rsk premum Fgure 7). The equty rsk premum s hgher n the constraned regon to compensate rsk-averse nvestors, but t falls quckly as the economy moves towards the pont ω 1, ω 2 ) = 0, 0). These changes n the nvestment opportunty set are drven by the constrant on portfolos, whch are represented n Fgure 8. As you can see, agent 1 s constraned over a large area of the state space. The portfolo weghts of agents 2 and 3 are knked at the nterface between regons where agent 1 s unconstraned and constraned. In the constraned regon, the portfolo weghts of unconstraned agents are steeper and portfolo weghts hgher than n the case of complete markets. Ths can be seen n Fgure 9, whch plots percentage devatons from the unconstraned equlbrum. Here we see that unconstraned agents hold substantally more of ther wealth n rsky assets than they would have n the unconstraned equlbrum. Ths pushes them closer to ther constrant. As agent 1 becomes more and more constraned, the nvestment opportuntes of agent 2 mprove, causng them to leverage up. Eventually they run nto ther constrant, creatng a sort of cascade. However ths ncrease n portfolo weghts does not translate drectly nto an ncrease n leverage. In the constraned regon, leverage s weakly lower than n the unconstraned equlbrum and exhbts non-monotonc and non-lnear dynamcs. Fgure 10 shows leverage n both levels and devatons from the unconstraned equlbrum. There are two peaks n leverage, one along the boundary where agent 1 becomes constraned and another along the boundary where agent 2 becomes constraned. In the ntermedate regon, leverage actually recovers back to ts unconstraned level, as you can see n Fgure 10b), ndcated by the dark red regon for low values of ω 1 and hgh values of ω 2. Ths mples that, even though agent 1 s constraned, agent two holds a suffcent amount of leverage to completely offset the reducton. At the same tme, they do not hold more leverage than necessary to push the economy back to the same amount of aggregate leverage that would preval wthout constrant. Fnally, the volatlty surface shows non-monotonc and non-lnear dynamcs. As we can see n Fgure 11, there s excess volatlty above the fundamental volatlty σ D. However, the constrant reduces ths because of a reducton n rsk sharng. We can thnk of the margn constrant as pushng the economy towards the autarkcal case, as ndvduals are unable to trade freely. In the lmt when there s no trade whatsoever, the volatlty of the asset prce s smply the volatlty of the underlyng dvdend. However, the constrant does not qute push the economy to ths pont, as ndvduals stll exchange n order to reman aganst ther constrant. All of these observatons pont to several emprcal tests, however the most apparent s 22

24 that of leverage. We can see that the cyclcalty of leverage s varyng wth other macroeconomc varables. Usng ths observaton we can thnk about a new way to consder leverage cyclcalty. 4. The Cyclcalty of Leverage Belefs drven leverage cycles are pro-cyclcal accordng to theory. Ths mplcaton s somewhat contradcted n several emprcal studes, ncludng Adran and Shn 2010b). In that paper the authors note that the leverage cycle s only pro-cyclcal for a partcular sector of the economy, asset broker/dealers. However, those authors plot leverage as a functon of total assets, whch produces a mechancal correlaton. Consder the defnton of fnancal leverage: Leverage = Labltes NetW ealth = Labltes Assets Labltes Increases n balance sheet assets produce a negatve correlaton between leverage and assets 8 Ang et al. 2011)). Fgure 12a) plots the rate of growth n leverage aganst the rate of growth n assets for all sectors over 1952Q1 to 2017Q1 as measured from the US Flow of Funds. As you can see, there s a clear negatve relatonshp. Consder nstead changes n GDP as a proxy for the busness cycle. Fgure 12b) plots the rate of growth n leverage for all sectors aganst the rate of growth n GDP over the same perod, agan from U.S. Flow of Funds data. The prevously clear negatve relatonshp has dsappeared, mplyng the leverage cycle s ambguous n ths sense. However, ths ambguty may smply be that there exsts some other explanatory varable whch drves the cyclcalty of leverage, n partcular preference heterogenety. One proxy for preference heterogenety s the prce-dvdend rato. As we saw n secton 2, asset prces wll be hgh relatve to dvdends and vce-versa when the margnal agent n the economy s less rsk averse. Fgure 13a) plots the growth rate n GDP aganst the prce of the S&P 500 dvded by GDP a measure of the prce/dvdend rato of the total economy). Indeed we see that there s substantal dsperson n ths measure. The prce/dvdend rato bunches towards the orgn as asset prces have been rsng over tme, but there s lttle evdence for a clear postve or negatve relatonshp wth GDP growth. For ths reason, we 8 However, ths makes the fact that Adran and Shn 2010b) fnd pro-cyclcal leverage cycles for broker/dealers all the more substantal of a fndng 23

25 Nonfnancal Corporatons Nonfnancal Prvate Busness HH s and Nonprofts All Sectors 1) 2) 3) 4) Intercept ) ) ) ) ln GDP ** *** ** ) ) ) ) S/D ** ) ) ) ) ln GDP S/D ** *** *** ) ) ) ) Standard errors n parentheses. : p 0.1, : p 0.05, : p 0.01 Table 2: Regresson results for dependent varable ln Lev for dfferent sectors of the economy. A postve and sgnfcant coeffcent on ln GDP mples procyclcalty, whle a negatve and sgnfcant coeffcent on the nteracton wth S/D mples counter-cyclcalty when the prce dvdend rato s hgh. Note: Varables are normalzed usng z-score. can consder the correlatons between these varables, captured by the followng regresson: ln Lev = α + β 1 ln GDP + β 2 ln GDP S D + β S 3 D The cyclcalty of the leverage cycle s then captured by the slope wth respect to the growth rate n GDP, that s ln GDP ln Lev = β 1 + β 2 S D The leverage cycle s pro- or counter-cyclcal as ths value s postve or negatve, respectvely Table 2 reports the results for several specfcatons, studyng dfferent subsamples of the economy. The results mply that the cyclcalty of leverage s not the same for all values of the prce-dvdend rato. Column 4 gves results for all sectors ncluded n the US Flow of Funds. Leverage growth s postvely correlated wth GDP growth when the prce dvdend rato s low. As asset prces rse the effect changes sgn and the correlaton becomes negatve. Changes n the prce-dvdend rato mply changes n the preferences of the margnal agent prcng rsky assets. When the prce-dvdend rato s low the margnal agent s rsk averse, whle when the prce-dvdend rato s hgh the margnal agent s more rsk neutral. Thus the leverage cycle s pro-cyclcal when rsk-averse agents domnate and counter-cyclcal when 24

26 Nonfnancal Corporatons Nonfnancal Prvate Busness HH s and Nonprofts All Sectors 1) 2) 3) 4) Intercept ) ) ) ) ln GDP *** ** *** *** ) ) ) ) r ) ) ) ) ln GDP r ** ** *** *** ) ) ) ) Standard errors n parentheses. : p 0.1, : p 0.05, : p 0.01 Table 3: Regresson results for dependent varable ln Lev for dfferent sectors of the economy. A negatve and sgnfcant coeffcent on ln GDP mples counter-cyclcalty, whle a postve and sgnfcant coeffcent on the nteracton wth r mples pro-cyclcalty when the nterest rate s hgh. As opposed to Table 2, r s hgh when the rsk averse agent domnates, exactly when the prce-dvdend rato s low. Note: Varables are normalzed usng z-score. rsk-neutral agents domnate. Ths result s farly robust to other measures of margnal preferences. One problem could be the heteroscedastcty exhbted by GDP growth over the prce/dvdend rato n Fgure 13a). Consder the rsk free rate as a proxy for the margnal agent, whch s plotted n Fgure 13b) aganst GDP growth. In ths case the dsperson of GDP growth s more unform over values of the nterest rate. Defne a smlar set of regressons as before,.e.: ln Lev = α + β 1 ln GDP + β 2 ln GDP r + β 3 r Agan the cyclcalty s captured by the slope wth respect to the growth rate n GDP: ln GDP ln Lev = β 1 + β 2 r In ths case we should expect the sgn to flp. The nterest rate s hgh when the margnal agent s rsk-averse and low when the margnal agent s rsk-neutral. Table 3 reports the results. Leverage growth co-moves postvely wth GDP growth and the nterest rate s hgh and negatvely when the nterest rate s low. Ths result agan mples that the cyclcalty of the leverage cycle depends n the same way as before on the preferences of the margnal agent. The regresson results hghlght how the cyclcalty of the leverage cycle relates to fnancal 25

27 varables and, n turn, preferences. Agents are lkely to be constraned when asset prces are low, producng a pro-cyclcal leverage cycle. Agents wll be far from ther constrant when asset prces are hgh, producng a counter-cyclcal the leverage cycle. Asset prce movements are explaned by changes n the margnal agent n the economy, as seen n secton Concluson In ths paper I ve shown how one can solve a model of preference heterogenety when agents face convex portfolo constrants. The results show how preference heterogenety and constrant can lead not only to cascade effects, but also hgh asset prces, hgh returns, and low nterest rates. I also show how leverage cycles can be both pro- or counter-cyclcal dependng on the underlyng assumptons of preference heterogenety and constrants. In addton I ve documented a new stylzed fact predcted by the model, namely that leverage s both pro- and counter-cyclcal dependng on the level of aggregate consumpton. Future work on ths topc could ntroduce a stochastc endowment and more general preferences. References Abbot, T. 2017). Heterogeneous preferences and general equlbrum n fnancal markets. Workng Paper. Adran, T. and Shn, H. S. 2010a). The changng nature of fnancal ntermedaton and the fnancal crss of Annu. Rev. Econ., 21): Adran, T. and Shn, H. S. 2010b). Lqudty and leverage. Journal of fnancal ntermedaton, 193): Ayagar, S. R. 1994). Unnsured dosyncratc rsk and aggregate savng. The Quarterly Journal of Economcs, pages Ang, A., Gorovyy, S., and Van Inwegen, G. B. 2011). Hedge fund leverage. Journal of Fnancal Economcs, 1021): Basak, S. and Cuoco, D. 1998). An equlbrum model wth restrcted stock market partcpaton. Revew of Fnancal Studes, 112): Bernanke, B. S., Gertler, M., and Glchrst, S. 1999). The fnancal accelerator n a quanttatve busness cycle framework. Handbook of macroeconomcs, 1:

28 Bhamra, H. S. and Uppal, R. 2014). Asset prces wth heterogenety n preferences and belefs. Revew of Fnancal Studes, 272): Brunnermeer, M. K. and Nagel, S. 2008). Do wealth fluctuatons generate tme-varyng rsk averson? mcro-evdence on ndvduals asset allocaton. The Amercan Economc Revew, 983): Brunnermeer, M. K. and Pedersen, L. H. 2009). Market lqudty and fundng lqudty. Revew of Fnancal studes, 226): Brunnermeer, M. K. and Sannkov, Y. 2014). A macroeconomc model wth a fnancal sector. The Amercan Economc Revew, 1042): Campbell, J. Y. and Cochrane, J. H. 1999). By force of habt: A consumpton-based explanaton of aggregate stock market behavor. The Journal of Poltcal Economy, 1072): Chabakaur, G. 2013). Dynamc equlbrum wth two stocks, heterogeneous nvestors, and portfolo constrants. Revew of Fnancal Studes, 2612): Chabakaur, G. 2015). Asset prcng wth heterogeneous preferences, belefs, and portfolo constrants. Journal of Monetary Economcs, 75: Chappor, P.-A., Gandh, A., Salané, B., and Salané, F. 2012). From aggregate bettng data to ndvdual rsk preferences. Chappor, P.-A. and Paella, M. 2011). Relatve rsk averson s constant: Evdence from panel data. Journal of the European Economc Assocaton, 96): Chrstensen, P. O., Larsen, K., and Munk, C. 2012). Equlbrum n securtes markets wth heterogeneous nvestors and unspanned ncome rsk. Journal of Economc Theory, 1473): Coen-Pran, D. 2004). Effects of dfferences n rsk averson on the dstrbuton of wealth. Macroeconomc Dynamcs, 805): Confortola, F., Cosso, A., and Fuhrman, M. 2017). Backward sdes and nfnte horzon stochastc optmal control. arxv preprnt arxv: Cozz, M. 2011). Rsk averson heterogenety, rsky jobs and wealth nequalty. Techncal report, Queen s Economcs Department Workng Paper. 27

29 Cuoco, D. 1997). Optmal consumpton and equlbrum prces wth portfolo constrants and stochastc ncome. Journal of Economc Theory, 721): Cuoco, D. and He, H. 1994). Dynamc equlbrum n nfnte-dmensonal economes wth ncomplete nformaton. Techncal report, Workng paper, Wharton School, Unversty of Pennsylvana. Cuoco, D., He, H., et al. 2001). Dynamc aggregaton and computaton of equlbra n fnte-dmensonal economes wth ncomplete fnancal markets. Annals of Economcs and Fnance, 22): Cvtanć, J., Joun, E., Malamud, S., and Napp, C. 2011). Fnancal markets equlbrum wth heterogeneous agents. Revew of Fnance, page rfr018. Cvtanć, J. and Karatzas, I. 1992). Convex dualty n constraned portfolo optmzaton. The Annals of Appled Probablty, pages Dumas, B. 1989). Two-person dynamc equlbrum n the captal market. Revew of Fnancal Studes, 22): Epsten, L. G., Farh, E., and Strzaleck, T. 2014). How much would you pay to resolve long-run rsk? The Amercan Economc Revew, 1049): Epsten, L. G. and Zn, S. E. 1989). Substtuton, rsk averson, and the temporal behavor of consumpton and asset returns: A theoretcal framework. Econometrca: Journal of the Econometrc Socety, pages Gârleanu, N. and Panageas, S. 2015). Young, old, conservatve and bold: The mplcatons of heterogenety and fnte lves for asset prcng. Journal of Poltcal Economy, 1233): Garleanu, N. and Pedersen, L. H. 2011). Margn-based asset prcng and devatons from the law of one prce. Revew of Fnancal Studes, 246): Geanakoplos, J. 1996). Promses promses. Geanakoplos, J. 2010). The leverage cycle. In NBER Macroeconomcs Annual 2009, Volume 24, pages Unversty of Chcago Press. Guvenen, F. 2006). Reconclng conflctng evdence on the elastcty of ntertemporal substtuton: A macroeconomc perspectve. Journal of Monetary Economcs, 537):

30 Guvenen, F. 2009). A parsmonous macroeconomc model for asset prcng. Econometrca, 776): Hallng, M., Yu, J., and Zechner, J. 2016). Leverage dynamcs over the busness cycle. Journal of Fnancal Economcs, 1221): Hardouvels, G. A. and Perstan, S. 1992). Margn requrements, speculatve tradng, and stock prce fluctuatons: The case of japan. The Quarterly Journal of Economcs, 1074): Hardouvels, G. A. and Theodossou, P. 2002). The asymmetrc relaton between ntal margn requrements and stock market volatlty across bull and bear markets. Revew of Fnancal Studes, 155): He, H. and Pages, H. F. 1993). Labor ncome, borrowng constrants, and equlbrum asset prces. Economc Theory, 34): He, Z. and Krshnamurthy, A. 2013). Intermedary asset prcng. The Amercan Economc Revew, 1032): Hugonner, J. 2012). Ratonal asset prcng bubbles and portfolo constrants. Journal of Economc Theory, 1476): Ish, H. 1989). On unqueness and exstence of vscosty solutons of fully nonlnear secondorder ellptc pde s. Communcatons on pure and appled mathematcs, 421): Karatzas, I., Lehoczky, J. P., and Shreve, S. E. 1987). Optmal portfolo and consumpton decsons for a small nvestor on a fnte horzon. SIAM journal on control and optmzaton, 256): Karatzas, I., Žtkovć, G., et al. 2003). Optmal consumpton from nvestment and random endowment n ncomplete semmartngale markets. The Annals of Probablty, 314): Kmball, M. S. 1990). Precautonary savng n the small and n the large. Econometrca: Journal of the Econometrc Socety, pages Kyotak, N. and Moore, J. 1997). Credt cycles. Journal of poltcal economy, 1052):

31 Kogan, L., Makarov, I., and Uppal, R. 2007). The equty rsk premum and the rskfree rate n an economy wth borrowng constrants. Mathematcs and Fnancal Economcs, 11):1 19. Korajczyk, R. A. and Levy, A. 2003). Captal structure choce: macroeconomc condtons and fnancal constrants. Journal of fnancal economcs, 681): Krusell, P. and Smth, Jr, A. A. 1998). Income and wealth heterogenety n the macroeconomy. Journal of Poltcal Economy, 1065): Longstaff, F. A. and Wang, J. 2012). Asset prcng and the credt market. Revew of Fnancal Studes, 2511): Lucas, R. E. 1978). Asset prces n an exchange economy. Econometrca: Journal of the Econometrc Socety, pages Mehra, R. and Prescott, E. C. 1985). The equty premum: A puzzle. Journal of monetary Economcs, 152): Merton, R. C. 1971). Optmum consumpton and portfolo rules n a contnuous-tme model. Journal of economc theory, 34): Preto, R. 2010). Dynamc equlbrum wth heterogeneous agents and rsk constrants. Radner, R. 1972). Exstence of equlbrum of plans, prces, and prce expectatons n a sequence of markets. Econometrca: Journal of the Econometrc Socety, pages Rytchkov, O. 2014). Asset prcng wth dynamc margn constrants. The Journal of Fnance, 691): Santos, T. and Verones, P. 2010). Habt formaton, the cross secton of stock returns and the cash-flow rsk puzzle. Journal of Fnancal Economcs, 982): Stgltz, J. E. and Wess, A. 1981). Credt ratonng n markets wth mperfect nformaton. The Amercan economc revew, 713): Wel, P. 1989). The equty premum puzzle and the rsk-free rate puzzle. Journal of Monetary Economcs, 243):

32 a) b) Fg. 5. Market prce of rsk n levels Fgure 5a)) and n devatons from complete markets Fgure 5b)). 31

33 a) b) Fg. 6. Asset prces n levels Fgure 6a)) and n devatons from complete markets Fgure 6b)). 32

34 a) b) Fg. 7. Asset prces n levels Fgure 7a)) and n devatons from complete markets Fgure 7b)). 33

35 a) b) c) Fg. 8. Portfolo weghts. On the left s a countour plot and the rght a surface plot of the same data. 34

36 a) b) c) Fg. 9. Portfolo weghts n percentage devatons from complete markets. 35

37 a) b) Fg. 10. Leverage n levels Fgure 10a)) and n devatons from complete markets Fgure 10b)). 36

38 a) b) Fg. 11. Volatlty n levels Fgure 11a)) and n devatons from complete markets Fgure 11b)). 37

39 a) b) Fg. 12. Growth rate n leverage plotted aganst the growth rate n assets Fgure 12a)) and aganst the growth rate n GDP Fgure 12b)) for all sectors. Source: FRB Flow of Funds Data. 38

40 a) b) Fg. 13. Growth rate n GDP plotted aganst the prce/dvdend rato Fgure 13a)), proxed by the prce of the S&P500 dvded by GDP, and aganst the rsk free rate Fgure 13b)), proxed by the yeld on constant maturty 10-year treasures. Source: FRB Flow of Funds Data and FRED. 39

41 Appendx A. Proofs Proof of Proposton 1. Take the market clearng condton n consumpton and dvde through by agent s consumpton c jt = D t c t = j c t j c D t = jt eρt Λ H t ) 1 γ j eρt Λ j H jt ) 1 γ j D t = ω t D t where ω t represents an ndvdual s consumpton weght and s gven by ω t = e ρt Λ H t ) 1 γ N j=1 eρt Λ j H jt ) 1 γ j Assume ndvdual consumpton follows an Itô process such that dc t c t = µ ct dt + σ ct dw t) 18) Apply Itô s lemma to Eq. 5) and solve for µ ct and σ ct µ ct = r t ρ + δ ν t ) γ γ γ 2 1 θ t + ν ) 2 t, σ ct) = 1 θ + ν ) t 2 σ t γ σ t Apply Itô s lemma to the market clearng condton for consumpton and match coeffcents to fnd µ D = N ω t µ ct, σ D = =1 N ω t σ ct =1 Now substtute the values for consumpton drft and dffuson and solve for the nterest rate and the market prce of rsk: θ t = 1 r t = 1 ω t γ ω t γ σ D 1 σ t µ D + ρ ω t ν t γ ω t γ ) ω t δ ν t ) γ θ t + ν ) 2 t ω t) γ 2 γ σ t Proof of Proposton 2. Apply Itô s lemma to ω t = c t D t and match coeffcents to fnd the dynamcs of consumpton weghts n Eq. 9) and Eq. 10). 40

42 Proof of Propostons 3 and 4. Assume there exsts a Markovan equlbrum n ω t = [ω 1t,..., ω N 1)t ]. In the ndvdual s fcttous fnancal market, they solve the followng optmzaton problem J t, x, ω) = max {c t,π t } u=t E s.t. dx t = t e ρu t) c1 γ u du 1 γ [X t r t + +δ ν t ) + π t σ t θ + ν t σ t dω t = µ ω dt + σ ω dw t δ ν t ) + ν t π t = 0 X t = x, ω t = ω )) c t ] dt + X t π t σ t dw t Then an ndvdual s Hamlton-Jacob-Bellman HJB) equaton wrtes 9 { 0 = max e ρt c1 γ + J [ t c,π 1 γ t + X r t + δ ν t ) + πσ t θ t + ν )) ] t Jt c σ t X N 1 N 1 + j=1 N 1 J t µ ωjt ω jt + π ω j j=1 2 J t σ ωjt σ ωkt ω j ω k ω j=1 j ω k k<j [ ]} + 1 X 2 π 2 σt 2 2 N 1 J t 2 X + σωjtω J t 2 j ωj 2 σ ωjt σ t ω j X 2 J t X ω j + j=1 19) subject to the transversalty condton E t J t 0 for all. Frst order condtons mply c = e ρt J t X 2 J t π = Xσ t X 2 ) 1 γ 20) ) [ 1 θ t + ν ) t Jt σ t Assume that the value functon s separable as ] N 1 X + 2 J t σ ωjt ω j X ω j j=1 21) J t, x, ω) = e ρt x1 γ V ω) γ 1 γ 22) 9 In the followng, the t subscrpt denotes dependence on the state. For all values other than J t, ths mples only ω. 41

43 Substtutng Eq. 22) nto Eqs. 20) and 21) gves c = x V ω) π = 1 γ σ t θ t + ν t σ t + γ N 1 V ω) j=1 ) V ω) σ ωjt ω j + ω j 23) 24) whch shows that V ω) s the wealth-consumpton rato as a functon of the vector of consumpton weghts. Defne as the gradent operator and use Eq. 11), then Eq. 24) rewrtes as π = 1 θ t + ν ) t σ + γ ω V T ω) γ σ t σ t V ω) as n Proposton 4. Next, substtute Eqs. 22) to 24) nto Eq. 19) and smplfy to fnd 0 =1 + 1 [ 1 γ 2 σt ωhv ω)σ ω + γ [ + 1 γ 1 γ )r t + δ ν t )) ρ + 1 γ 2γ θ t + ν ) ] t σω T + µ T ω V ω) σ t θ t + ν ) ] 2 t V ω) σ t Where µ ω and σ ω are as n Proposton 2, and where H represents the Hessan operator. The boundary condtons are gven by recognzng that the lmts n ω {0, 1} s an economy where agent has zero weght, whle the remanng agents determne prces. Proof of Proposton 5. Defne the prce-dvdend rato as a functon of consumpton weghts: Sω) = S t = St D t. Takng the market clearng condton for wealth 25) S t = X t S t D t = X t D t = X t c t c t D t = V ω)ω = Sω) To fnd volatlty, apply Itô s lemma to D t V ω t )ω t = S t and match coeffcents to fnd the expresson n Eq. 15). Proof of Proposton 6. For a homogeneous margn constrant, ν t 0 and m 0, thus ν t m 0 Cvtanć and Karatzas 1992); Chabakaur 2015)). Addtonally, π t m. Substtutng the soluton for π t from Eq. 14) nto the latter nequalty and recognzng that, by the Kuhn-Tucker condtons at least one of the nequaltes holds wth equalty gves the result. Proof of Clam 1. Ths proof shows that the present settng satsfes the assumptons neces- 42

44 sary to apply Proposton 5.1 and Theorem 5.1 from Confortola et al. 2017), namely that the problem admts a dynamc programmng representaton and that the soluton n the vscosty sense) to the HJB n Eq. 19) s ndeed the value functon. The proof proceeds n three steps: frst the state s defned as a Markov dffuson, second the assumptons from Confortola et al. 2017) are verfed, and thrd the assumptons from Ish 1989) are verfed, gvng unqueness of the vscosty soluton. Markovan Defne Y t = [X t, ω 1t,..., ω N 1)t ] T R + N 1 = X as the state vector of an ndvdual and α t = [c t, π t ] T R + Π = A as the control vector. By Propostons 1 to 6 the state vector s Markovan and has controlled dynamcs dy t = by t, α t )dt + σy t, α t )dw t Bellman Prncple and Exstence of Vscosty Soluton Defne the felcty functon fy, a) = u c) and notce that ths s not a functon of the state and only a functon of one of the controls. The followng assumptons must be verfed: Assumpton A.1. The dynamcs of wealth, the dynamcs of consumpton weghts, and the utlty functon must be contnuous. Assumpton A.2. The dynamcs of wealth must be jontly Lpschtz contnuous n the state varable. Assumpton A.3. There must exst some constants M > 0 and r 0 such that fy, a) M1 + y r ) for all y R N and all a A. Assumpton A.4. ρ > ρ, where ρ = 0 f r from assumpton A.3 s zero, otherwse f r > 0, ρ > 0 s such that E [ sup Y s s [0,t] ] Ce ρt 1 + x r ) for some constant C 0, wth ρ and C ndependent of t, α, and x. Assumpton A.5. fx, a) s contnuous n x unformly wth respect to a. Assumpton A.5 s trvally satsfed as the utlty functon s not a functon of the state, but only of the control. 43

45 Assumpton A.3 s easly shown to be satsfed snce the utlty functon s of power form: c 1 γ 1 γ M1 + y r ) takng r = 0 we have c 1 γ 21 γ ) M whch s satsfed n fnte t by takng M max t {D t } 1 γ /21 γ )) n any fnte t, thanks to market clearng. In the lmt when t, we have D t gong to, a.s. In ths case, the admssble set goes to a sngleton. Assumpton A.4 s satsfed for all non-zero rates of tme preference, gven we took r = 0 above. Assumpton A.1 s satsfed for the utlty functon. For the dynamcs of wealth, the functons are contnuous up to ν ) and V ). If we assume that V ) s once contnuously dfferentable, then we satsfy assumpton A.1. Assumpton A.2 Is satsfed f V ) has bounded frst dervatve. Unqueness of Vscosty Soluton To be completed) Outlne of proof: accordng to Ish 1989), f the PDE gven n Eq. 12) s unform ellptc, f sub- and super-solutons have exponental growth, and under the above assumptons on contnuty of the dynamcs of the state, then the vscosty soluton s unque. Appendx B. Numercal Method In ths appendx I dscuss the numercal soluton of the problem of three preference types under margn constrants. Ths problem has several dffcult features whch make t challengng from a numercal perspectve. Frst, the system of PDE s n?? represents a hghly non-lnear system, as the coeffcents depend n a non-trval way on the soluton, as well as on the Jacoban. In addton, the coeffcents are not smooth at the pont where the constrant bnds. Fnally, the state space s the 2 dmensonal smplex, makng fnte dfference tedous. 44

46 B.1. The Dscretzed State Space Consder frst the state space when N = 3. In ths case ω {x [0, 1] 2 x 1}. Fgure 14a) plots ths space as the shaded regon, smply the lower trangle of the [0, 1] square n the frst quandrant. To solve the PDE wth fnte dfference we must dscretze ths a) b) Fg. 14. The two dmensonal standard smplex Fgure 14a)) and ts dscretzed verson Fgure 14b)) wth K = 10 ponts along each axs. Black ponts are nteror ponts and whte ponts are boundary ponts space. I take the conventon of specfyng a dscretzaton by the number K of ponts along each axs. Fgure 14b) gves an example of such a dscretzaton when K = 10. If we defne L as the number of ponts n the state space we have L = K K + 1)/2 ponts. Notce that along each edge of the smplex we have boundary condtons, so the number of nteror ponts s K 1)K 1)/2. B.2. Boundary Vertces The boundary condtons n the state space are very non-trval. At each extreme pont n the smplex, a sngle agent domnates. That s ω 1, ω 2 ) = 1, 0) Agent 1 domnates. ω 1, ω 2 ) = 0, 1) Agent 2 domnates. ω 1, ω 2 ) = 0, 0) Agent 3 domnates. 45

47 For each ω vj n ths set of vertces subscrpt denotes vertex wth domnant agent j), the domnant agent s prce domnates. We have the followng aggregate varables θω vj ) = σ D γ j rω vj ) = ρ + µ D γ j γ j 1 + γ j ) σ2 D 2 σω vj ) = σ D In addton, ndvduals may be constraned on the vertces. adjustments satsfy It can be shown that ther ν ω vj ) = mn { 0, m γ γ j )σ 2 D } Fnally, the wealth/consumpton ratos on these vertces are gven by V ω vj ) = γ ) σd +ν ω vj )/σ D) 2 ρ 1 γ ) 2γ + rω vj ) m ν ω vj ) B.3. Boundary Edges Along a boundary edge, we are n a case where one agent has zero consumpton weght and the other two agents have weght that vares. There are three cases: ω 1, ω 2 ) {x, y) x [0, 1], y = 0} ω 1, ω 2 ) {x, y) x = 0, y [0, 1]} ω 1, ω 2 ) {x, y) [0, 1] 2 x + y = 1} If we take the state varable to be the consumpton weght of the ndvdual wth the lowest ndex j and who has non-zero weght along the edge, the PDEs become ODEs: 1 γ [ [ 1 γ γ 1 γ )rω j ) m ν ω j )) ρ + 1 γ 2γ θω j ) + ν ω j ) σω j ) θω j ) + ν ) ] 2 ω j ) V ω j )+ σω j ) ) σ ωj ω j + µ ωj ω j ] V ω j ) σ2 ωjv ω j ) + 1 = 0 The vertex values n secton B.2 gve boundary condtons for solvng the two agent problem. Already ths ODE problem s hghly non-lnear. To solve t I use an mplct scheme and Pcard teraton. For the mplct method, we can add a tme dervatve term and consder 26) 46

48 the long run level of the wealth/consumpton rato to be t, the length of the dscrete tme step. Dscretze the state space along the edge nto P ponts ω p j = j/p j {1,..., P }. Denote V t, ω p t,p j ) as V. In addton, defne the coeffcents n secton B.3 such that t V t,p + a ω p j, V t,p )V t,p + bω p j, V t,p ) ω V t,p + cω p j, V t,p ) 2 ωωv t,p + 1 = 0 To carry out Pcard teraton, ntalze the soluton at the termnal value, then evaluate the coeffcents usng the current soluton guess and treat the dervatves as unknowns. Usng the followng second-order-accurate central dfference schemes: t V t,p ω V t,p 2 ωωv t,p V t+1,p V t,p+1 V t,p+1 t V t,p V t,p 1 2h 2V t,p h 2 + V t,p 1 where h = 1/P, and usng a smlar superscrptng scheme for the coeffcents, the dscretzed scheme can be rearranged as [ c t+1,p h 2 ] bt+1,p V t,p 1 + [a t+1,p 2ct+1,p 1 ] [ c V t,p 1 t+1,p 2h h bt+1,p t h 2 2h Ths system of lnear equatons can be wrtten as = ] V t,p 1 ] [1 + V t+1,p t Ax = b where A s trdagonal. At each tme step ths system s solved and the algorthm stops when two consecutve steps are suffcently close. An example of a soluton to the two agent problem s gven n?? 47

49 B.4. Full State Space Gven the numercal soluton n sectons B.2 and B.3, we have boundary values and can now turn to the full PDE. For two agents the PDE n Proposton 3 becomes [ 1 1 γ )rω) + δ ν ω))) ρ + 1 γ θω) + ν ) ] 2 ω) V ω)+ γ 2γ σω) [ 1 γ θω) + ν ) ] ω) σ ω1 ω)ω 1 + µ ω1 ω)ω 1 ω1 V ω)+ γ σω) [ 1 γ θω) + ν ) ] ω) σ ω2 ω)ω 2 + µ ω2 ω)ω 2 ω2 V ω)+ γ σω) 1 [ σ 2 2 ω1 ω1 2 ω 2 1 ω 1 V ω) + σω2ω ω 2 2 ω 2 V ω) + 2σ ω1 σ ω2 ω 1 ω 2 ω 2 1 ω 2 V ω) ] + 1 = 0 As stated n secton B.1, we dscretze each edge nto K ponts, gvng a total of KK + 1)/2 ponts n the state space. If we denote ω jk = [ω j 1, ω k 2] T the j, k) th pont n the state space and f we use a smlar superscrptng scheme as n secton B.3, we can use the followng set of second-order-accurate central dfference schemes: 27) t V t,j,k ω1 V t,j,k ω2 V t,j,k 2 ω 1 ω 1 V t,j,k 2 ω 2 ω 2 V t,j,k 2 ω 1 ω 2 V t,j,k V t+1,j,k V t,j+1,k V t,j,k+1 V t,j+1,k V t,j,k+1 t V t,j+1,k+1 V t,j,k V t,j 1,k 2h V t,j,k 1 2h 2V t,j,k h 2 2V t,j,k h 2 + V t,j 1,k + V t,j,k 1 V t,j 1,k+1 V t,j+1,k 1 4h 2 + V t,j 1,k 1 If n addton we use the same Pcard teraton scheme, we can dscretze the equaton as a t+1,j,k,1 V t,j,k + a t+1,j,k,2 V t,j+1,k + a t+1,j,k,3 V t,j 1,k + a t+1,j,k,4 V t,j,k+1 + a t+1,j,k,5 V t,j,k 1 + a t+1,j,k,6 V t,j+1,k+1 + a t+1,j,k,7 V t,j+1,k 1 + a t+1,j,k,8 V t,j 1,k+1 + a t+1,j,k,9 V t,j 1,k 1 = b t+1,j,k 48

50 where the coeffcents are gven by a t,j,k,1 = 1 γ σt+1,j,k ω1 h 2 a t,j,k,2 = σt+1,j,k ω1 a t,j,k,3 = σt+1,j,k ω1 a t,j,k,4 = σt+1,j,k ω2 a t,j,k,5 = σt+1,j,k ω2 a t,j,k,6 = σt+1,j,k ω1,7 = σt+1,j,k ω1 a t,j,k,8 = σt+1,j,k ω1 a t,j,k a t,j,k,9 = σt+1,j,k ω1 b t,j,k 1 γ )r t+1,j,k m ν t+1,j,k ) ρ + 1 γ 2γ ω j,k 1 ) 2 ω j,k 1 ) h 2 σt+1,j,k ω2 h 2 2h ω j,k 1 ) 2 1 2h 2 2h ω j,k 2 ) h 2 2h ω j,k 2 ) 2 1 2h 2 σ t+1,j,k ω2 4h 2 σ t+1,j,k ω2 4h 2 σ t+1,j,k ω2 4h 2 σ t+1,j,k ω2 4h 2 = 1 V t+1,j,k t 2h ω j,k 1 ω j,k 2 ω j,k 1 ω j,k 2 ω j,k 1 ω j,k 2 ω j,k 1 ω j,k 2 ω j,k 2 ) 2 [ 1 γ γ [ 1 γ γ [ 1 γ γ [ 1 γ γ 1 t θ t+1,j,k + νt+1,j,k σ t+1,j,k θ t+1,j,k + νt+1,j,k σ t+1,j,k θ t+1,j,k + νt+1,j,k σ t+1,j,k θ t+1,j,k + νt+1,j,k σ t+1,j,k ) ) ) ) θ t+1,j,k + νt+1,j,k σ t+1,j,k σ t+1,j,k ω1 ω j,k σ t+1,j,k ω1 ω j,k σ t+1,j,k ω2 ω j,k σ t+1,j,k ω2 ω j,k ) µ t+1,j,k ω1 1 + µ t+1,j,k ω1 2 + µ t+1,j,k ω2 2 + µ t+1,j,k ω2 ω j,k 1 ω j,k 1 ω j,k 2 ω j,k 2 ] ] ] ] B.5. The Dagonal Boundary One ssue s that across the dagonal boundary the fnte dfference approxmaton wll reach beyond the state space. To deal wth ths ssue, we can defne an addtonal set of boundary ponts along the dagonal boundary and use a specal fnte dfference scheme to approxmate ths cross partal. Snce 2 xyfx, y) = x y fx, y)), we can use a central dfference n one drecton and a backward dfference n the other. Ths wll allow us to avod crossng the boundary. Ths approxmaton s gven by 2 xyfx, y) 1 2h 2 [f x,y+h f x h,y+h f x,y h + f x h,y h ] 28) 49

51 Fg. 15. When K = 4 we have the mnmum number of ponts to have an nteror pont. When we attempt to approxmate the cross partal at ths pont the fnte dfference approxmaton attempts to reach across the dagonal curved arrow). To deal wth ths we can ntroduce a partcular dfference approxmaton Eq. 28)), whch nstead makes use of the ponts whch exst n the state space straght arrows). Usng ths dscretzaton for the cross-partal at the dagonal boundary, we have the followng dscretzaton at each ndex j, K j): If n addton we use the same Pcard teraton scheme, we can dscretze the equaton as ã t+1,j,k j,1 V t,j,k j + ã t+1,j,k j,2 V t,j+1,k j + ã t+1,j,k j,3 V t,j 1,K j + ã t+1,j,k j,4 V t,j,k j+1 + ã t+1,j,k j,5 V t,j,k j 1 + ã t+1,j,k j,6 V t,j 1,K j 1 + ã t+1,j,k j,7 V t,j 1,K j+1 = b t+1,j,k 50