Online Supplementary Materials for Banking and Shadow Banking

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1 Online Supplemenary Maerials for Banking and Sadow Banking Ji Huang Naional Universiy of Singapore February 2, 28 Tis noe collecs supplemenary maerials for Banking and Sadow Banking ereafer BSB. Tis noe is no o be publised. Secion conains e calibraion of parameers and e source of aggregae securiizaion daa a BSB uses. Secion 2 develops a varian of e baseline model in wic bankers can issue ouside equiy o some exen. In Secion 3, we focus on an exension were ouseolds ave Epsein-Zin preferences. Secion 4 concenraes on a varian in wic regular banks face e capial requiremen consrain. Proofs and figures are in e end of e noe. Calibraion and Daa We se e ime discoun facor o 3% o mac e real ineres rae esimaed by Campbell and Cocrane 999. Bankers reiremen rae χ is se a 5% o arge e average Sarpe raio. We se bankers produciviy a 22.5% so a e average invesmen-o-capial raio is close o % He and Krisnamury, 22. Te produciviy of less-producive ouseolds is cosen a % o mac e fac a e Sarpe raio during e 27-9 financial crisis was approximaely 5 imes e average level He and Krisnamury, 22. Coices of e depreciaion rae and e capial adjusmen cos φ are sandard in e macroeconomic lieraure Crisiano, Eicenbaum and Evans, 25. We se e Poisson sock parameers o arge e condiional volailiy of e grow rae of bankers weal in disressed periods and in non-disressed periods. Te disressed periods are defined as periods wi e lowes 33% Sarpe raios. Te regulaion parameer, ax rae τ, is se a 3% o arge e average leverage of e enire banking secor. We se e Conac Deails: AS2 #6-2, Ars Link, Singapore jiuang@nus.edu.sg

2 Table : Momens Momen Model Targe Source average Sarpe raio 32.3% 4% Wacer 23 iges Sarpe raio average Sarpe raio 6. 5 He and Krisnamury 22 average invesmen capial raio.7% % He and Krisnamury 22 average raio of raio of securiizaion 25.% 25.% securiizaion in ird quarer of 26 bankers overall leverage He and Krisnamury 22 volailiy of bankers weal grow rae in disress periods % 3.5% He and Krisnamury 22 in non-disress periods 8.2% 7.5% He and Krisnamury 22 We use e densiy of e saionary disribuion o calculae all momens. 2 Te disress periods are ose wi iges 33% Sarpe raio. inensiy wi wic bankers can re-access sadow banking afer defaul a 6% o arge e raio of securiizaion by non-agency issuers in e ird quarer of 26. Table 2: Deails of Securiizaion Daa Ousanding Securiized Home Morgages FL FL Mulifamily Residenial Morgages FL FL Commercial Morgages FL FL Commercial and Indusrial FL FL Loans FL FL Consumer Credi FL5366 FL67366 Becuase iem FL is no available now, we use e raio of securiizaion calculaed by Louskina 2 o esimae e ousanding commercial and indusrial loans. Securiizaion. We follow Louskina 2 o compue e raio of securiizaion. Te difference is a we focus on securiizaion done by non-agency securiy issuers. All daa are drawn from e Flow of Funds Accouns of e Unied Saes. Tere are 5 loan caegories. Te deails of iems for eac caegory are lised in Table 2. 2

3 2 Risky Sadow Bank Deb In is secion, we consider an exension of e baseline model in wic bankers can issue ouside equiy up o ε proporion of eir banks ousanding equiy. For sadow banks, is implies a invesors of a sadow bank are willing o bear ε proporion of asse losses a occur o e sadow bank. Le R e denoe e reurn of ouside equiy and r s e reurn of sadow bank deb. A banker s dynamic budge consrain is dw = W R + ε W R R e + S R r τ + S R r s d ε W + W π c d ε ε + S + D S x q dn. Te regular bank raise εw /ε ouside equiy. If a Poisson sock is e economy, e banker only bears ε proporion of e loss o e regular bank. Wi respec o sadow banking, e regular bank exends implici guaranees a cover ε proporion of e loss o e sadow bank. Le E denoe a ouseold s olding of a regular bank s ouside equiy and O is olding of sadow bank deb. Te dynamic budge consrain for a ouseold is dw = W r + S R r + E R e r + O r s r c S + E + εs + εo x q dn, were s is e raio of regular bank deb o inside bank equiy. If a ouseold conribues dollar as ouside equiy, banker will mac ε / ε dollar as inside equiy. Taking leverage ino accoun, dollar conribuion of ouside equiy raises e size of a regular bank by +εs / ε. Te risk exposure of dollar ouside equiy is + εs x q d if a Poisson sock is e economy. Houseolds will ake {r, R, R e, r s, } as given and coose {S, E, O } o maximize eir expeced life-ime uiliy. Firs-order condiions wi respec o E and O are R e r = λ + εs x q, r s r = λ εx q. If we plug in e expressions of R e and r s back o a banker s dynamic budge consrain, en we derive e dynamic budge consrain lised in BSB. Nex, we derive e opimaliy condiions of bankers. Given e dynamic budge consrain 3

4 lised in BSB, firs order condiions are R r τ ελx q R r ελx q ελx q + εs + s, wi equaliy if s > xq ελx q + εs + s, wi equaliy if s > xq Since ouseolds are risk-neural and bankers are risk-averse, bankers are willing o ake iger leverage if ey can sare aggregae risks wi ouseolds. Te counerpar of e enforcemen consrain 5 in e case is s R r τ + s R r εs + s λx q + λ ln + εs + s x q + λĥ s R r τ + s R r ε s + s λx q + λ ln + ε s x q + λĥd. We can simplify e enforcemen consrain and derive e maximum leverage of sadow banking lised in BSB by following an argumen a is similar o a in e Secion 2..4 in BSB. Te procedure of solving for e equilibrium of is varian is e same as i is for e baseline model in BSB. Deailed discussion of key properies of is varian is conained in BSB as well. 3 Houseolds wi Epsein-Zin Preference In is secion, we consider a varian of e baseline model were ouseolds ave Epsein-Zin preferences wi e ime discoun rae, e relaive risk-aversion coefficien γ, and e elasiciy of ineremporal subsiuion b. were and In e modified model, e ouseold cooses {c, S, n, } o maximize [ U = E f c s, Us f { c, U = c b b γ U γ b γ U s = E s [ subjec o e dynamic budge consrain 6 in BSB. s ds ], γ U } f ] c v, Uv dv, for s >, 4

5 In solving for e equilibrium of e modified model, we need o caracerize e opimal consumpion and porfolio coices of ouseolds. To do so, we conjecure a e coninuaion value of a ouseold wi ne wor W as e following funcional form V = V ζ, W ζ W γ, γ were ζ follows dζ = ζ µ ζ d ζ ˆζ dn. ζ is inerpreed as e coninuaion value muliplier of a ouseold s ne wor as e funcion V ζ, W is omogeneous of degree γ wi respec o W. Given our conjecure, e Hamilon- Jacobi-Bellman equaion of e ouseold s dynamic programming problem is were D c,s V ζ, W = W r + S { = max f c c,s, V + D c,s V ζ, W }, R r c ζ γ W γ + µ ζ ζ W γ ˆζ W + λ S x q γ γ We summarize e key resuls of e problem in e following proposiion. ζ W γ Proposiion Eac ouseold s opimal consumpion weig { c, }, opimal porfolio weig { s, n, }, and e process {ζ, } saisfy γ. c b = ζ b, 2 R r λx q γ ˆζ s x q γ, wi equaliy if s >, 3 ζ = c b bζ b b + r + s ˆζ s R r c + µ ζ x q γ + λ γ γ 4 were c = c / W and s = S / W. Tere are a few marke clearing condiions in e modified model a are differen from ose in e baseline model. Firs, e risk-free rae is no consan, wic insead is joinly deermined 5

6 by e opimal porfolio coices of bo bankers and ouseolds equaion 3 and e dynamics of e coninuaion value muliplier {ζ, } equaion 4. Second, e consumpion good marke does no clear auomaically in e modified model since ouseolds are risk-averse. Given ouseolds opimal consumpion coice condiion 2, we ave e marke clearing condiion for e consumpion goods aψ + a ψ gι = ω + b b ζ b ω q. As in BSB, we focus on e Markov equilibrium of e modified model were sadow banking exiss. Te coice of parameer values is a = 4%, γ = 2, b =.5, χ =., a =.225, a =., δ = %, φ = 3, λ =, x = 4%, τ =.2%, and ξ = 5%. Figures 3 and 4 in Secion 6 sow a resuls found in e baseline model survive in e modified model. In wa follows, we sudy e welfare implicaion of bank regulaion of e modified model. As in BSB, we suppose a e amoun of pysical capial in period equals one and ere are a represenaive banker and a represenaive ouseold in e economy. Te welfare pair of e represenaive ouseold and banker is ζ ω q γ γ, lnω q +, were ζ is e coninuaion muliplier of e represenaion ouseold in period. Differen from e baseline model, e equilibrium risk-free rae in e modified model depends on e credi demand and supply. If e regulaory auoriy sars regulaing regular banking and sadow banking is sill unsusainable, e decrease in e banker s credi demand lowers e risk-free rae Panel d in Figure 5, wic urs e ouseold Panel b in Figure 5. Naurally, e banker s welfare improves because of e low volailiy of er weal and e low borrowing cos. Overall, imposing ax on regular banking sill improves e sum of wo represenaive agens welfare wen sadow banking does no emerge Panel c in Figure 5. Wen sadow banking is susainable, igening bank regulaion sars o improve e ouseold s welfare. Tere are wo underlying forces. Firs, e increased credi demand from sadow banks puses up e risk-free rae, wic benefis e ouseold. Panel d in Figure 5 sows a is effec is so large a e average risk-free rae in an economy wi a posiive ax rae could be iger an i is in an economy wiou any ax. Te underlying inuiion is a if e banker as o pay a ig ax rae for regular banking er willingness o pay a relaively ig ineres rae for sadow banking mus be ig as well. 6

7 Te second force is a as e sadow banking secor expands e banker olds more fracion of capial goods in e economy and us e ouseold s exposure o e aggregae risk declines. Panel d in Figure 6 indicaes a e volailiy of e ouseold s weal declines as e ax rae increases unil i is 4%. Overall, if we consider e sum of e wo agen s welfare, we observe a srengening bank regulaion can raise e oal welfare of e economy. Nevereless, wen regulaion is oo ig, igening regulaion makes social welfare worse off. Tis is primarily e consequence of e increased financial insabiliy a e expansion of e sadow banking secor causes. In summary, is secion igligs novel cannels, roug wic igening regulaion of regular banking improves e ouseold s welfare. In paricular, srengening bank regulaion elps e sadow banking secor expand, and e consequenial increase in credi demand raises e reurn for e ouseold o supply funds and lowers e ouseold s exposure o e aggregae risk. 4 Quaniy Conrol In is secion, we invesigae a varian of e baseline model in BSB, in wic e regular banking secor is subjec o a quaniy conrol. In paricular, we consider e capial-requiremen consrain, wic imposes an upper bound s for a regular bank s deb-o-equiy raio. Wi e price conrol replaced by e quaniy conrol, a banker s dynamic budge consrain becomes dw = W R + S + S R r c d W + S + S x q dn. In addiion o e leverage consrain for sadow banking 4, e banker in e modified model faces e capial-requiremen consrain S sw. Similar canges apply o bankers wo canno access sadow banking due o defaul. We firs focus on e dynamics of endogenous variables and en move o e regulaory smile resul of e quaniy-conrol model. A number of endogenous variables ave dynamics similar o e baseline model Panels a-e in Figure 7. However, e leverage dynamics of sadow banking cange drasically. Tis is e consequence of e fac a wen bankers sare of weal is small e excess reurn is ig and e incenives o build up leverage are srong. In ese saes, i is exremely cosly o defaul on sadow bank deb because regular banking only allows for considerably low leverage. Terefore, wen bankers sare of weal is small, e enforcemen problem is no severe and e leverage of sadow banking is ig. Tis propery is absen in e 7

8 baseline model because bankers do no face a binding leverage consrain for regular banking. Te counerfacual resul of sadow banking being couner-cyclical is primarily e consequence of e simplificaion a bankers also play e role of producers. In our model, wen e capial misallocaion is already severe in recessions, e ime-invarian capial raio requiremens acually prevens more producive agens from raising exernal credi, wic is apparenly unwise and in e opposie of improving social welfare. In realiy, wa we ofen observe is a governmens uilize all possible policy insrumens in downurns o encourage banks credi supply o e real secor. Figure 8 sows a e regulaory smile resul coninues o old in e modified model wi quaniy conrol. Very lenien bank regulaion comes wi e low leverage of sadow banking and ig financial insabiliy. As bank regulaion igens i.e., e maximum leverage of regular banking s declines, financial insabiliy iniially diminises. However, if e regulaion is so ig a e sadow banking secor becomes sizeable, iger regulaion causes iger financial insabiliy. 5 Proofs Proof of Lemma 2 in BSB. Now suppose a ime, e banker s weal W is negaive. Te law of moion for e banker s weal is dw = W R + S R r + S R r c d W x q + S x q + S x q dn Given a fixed ime T, we can consruc a new measure under wic R s r s = λ x q s for eac ime s beween and T. Under is new measure, T T Ẽ [W T exp r u du + c s exp s ] r u du ds W < Suppose e banker reires a e sopping ime S wi posiive weal W S. Afer e banker reires, er weal evolves as dw S+u = W S+u r S+u du. 8

9 S+s Her weal in period S + s is W S exp S r u du s. I is easy o see a [ lim E S exp s S+s S ] r u du W S+s =. Tus, if T is large enoug, E [W T exp ] T r u du could be arbirarily small. Since W <, e consumpion of e banker mus be negaive a some poin beween and T wi a sricly posiive probabiliy. Since e banker as logarim preference, e banker s expeced lifeime discouned uiliy a ime mus be negaive infiniy. Terefore, we sow a i is never opimal for e banker o ave negaive weal and a a banker s overall leverage mus ave an upper bound. Proof of Proposiion in BSB. Wiou loss of generaliy, we focus on a banker wi weal W in period and explicily express er coninuaion value in differen cases. We sar wi e case a e banker is reired. Since logarimic agens only consumer fracion of eir weal, e grow rae of er weal is r +v in period +v. Hence, e banker s weal in period + u will be W exp u r +v dv. Te banker s coninuaion value a ime is exp u ln W + = ln W ln + + = ln W ln + + wic is denoed by ln W / + r. u exp u r +v dv du u r +v dv du exp v r +v dv, We use e same idea o express e coninuaion value of a banker wo can access sadow banking. Given e banker s opimal porfolio coices s +u, s +u, if se does no reire in period + u, er weal in period + u, W +u is W +u W exp Since E [N +d N ] = λd, we rewrie W +u u R+v + s +v R +v r +v + s +v R +v r +v dv + u ln + s +v + s +v x q. +v dn+v u W +u = W exp G +v dv, were G +v R +v + s +v R +v r +v + s +v R +v r +v λ ln + s +v + s +v x q +v. 9

10 Le + T denoe e sopping ime a e banker reires. Her coninuaion value a ime is E [ T =E [ T =E [ T = lnw = lnw = ln W exp u lnw + u exp u lnw + ln + lnw+t G +v dv du + exp T u exp u lnw + ln + G +u + E [ T + E [ + ln + χ + E G +v dv du + exp T lnw du + exp T exp u ln + G +u du + exp T r +T T χ exp χt [ exp u exp + χu G+u + r +T ] lnw + ] + r +T ] T ln + G +u du + exp T r +T ] + χ r +u du G +v dv + r +T ] dt wic we denoe as ln W / +. We ave used inegraion by pars muliple imes in e above derivaion. Finally, we consider e case a e banker wo canno use sadow banking bu obain suc opporuniy a inensiy ξ. Le s denoe er opimal porfolio coice, wic saisfies ] R r τ = λx q + s x q, 5 and T ξ e sopping wen e banker obain e access o sadow banking. Her coninuaion value a ime is mint,tξ exp u ln W + u R +v + s +v R +v r +v τ +v + s +v τ +v dv du E + mint,t ξ exp u u ln + s +v x q +v dn+v du ln W +Tξ + exp T ξ + mintξ,t=t ξ + exp T lnw+t + r +T mintξ,t=t T u =E exp u ln W + G +v du + exp T ln Ŵ+T + r +T [ mint,tξ u E exp u s +vτ +v dv du + exp min T, T ξ ] mint,tξ s +vτ +v dv = ln W / + H,

11 were s +v τ +v is e ax rebae o eac banker in e economy and Ŵ +T W exp u R+v + s +v R +v r +v + s +v R +v r +v dv + u ln + s +v + s +v x q. +v dn+v Te firs equaion comes from e condiion a s + s = s. Nex, we verify a is condiion always olds. If s >, e firs-order equaion 3 in BSB and equaion 5 implies s + s = s. If s =, en τ = and e firs-order equaion 4 in BSB and equaion 5 also give rise o s + s = s. [ mint,tξ u H = E exp u s +vτ +v dv du + exp min T, T ξ [ = E exp + χ + ξ u τ ] +u s +udu mint,tξ s +vτ +v dv ] Proof of Teorem in BSB. We will apply e conracion mapping eorem o sow e uniqueness of e soluion H ω =. Firs, we define a complee meric space. Since e sae variable ω is beween and ω, we focus on e space B, ω] of bounded coninuous funcions :, ω] R under sup norm. Teorem 3. in Sokey e al. 989 implies a B, ω] is a complee meric space. We will use Blackwell s sufficien condiions o sow Γ is a conracion mapping. Suppose bo, B, ω] and ω ω, for all ω, ω], since s ω = λhˆω Rω r τω, all porfolio coices permied under are feasible under. Hence, Γ Γ, for all ω, ω]. Nex, we need o sow a ere exiss a posiive consan β < suc a Γ + v Γ + βv, for all B, ω], v, ω, ω]. Consider Γ + v [ω] = E [ exp + ξ + χ u min{τ, τs u} ] s udu ω = ω, were s ω λˆω Rω r τω,

12 and s, s are porfolio weigs of a banker in e equilibrium of a ypoeical economy wi exogenous. Since e lower bound of is zero, en s ω = = = λˆω + v Rω r τω = λˆω Rω r τω + λv Rω r τω λˆω Rω r τω + v + sω + s ωx q ω x q ω λˆω Rω r τω + v x q ω + sω + s ω λˆω Rω r τω + v x Wi e assisance of above inequaliy, we derive a Γ + v [ω] Γ[ω] + E [ Γ[ω] + τ + ξ + χ x v. If τ < + ξ + χ x, Γ is a conracion mapping. exp + ξ + χ u v ] x τdu ω = ω 6 Figures 3 densiy of saionary disribuion Figure : e densiy of saionary disribuion. For parameer values, see Secion 2.3 in BSB. 2

13 s ψ fracion of capial eld by bankers a leverage of sadow banking d q x q price of pysical capial b invesmen risk e R r τ s + s overall leverage c profiabiliy of banking f Figure 2: Tis figure presens e fracion of pysical capial eld by bankers ψ, e price of pysical capial q, banker s overall leverage s + s, e leverage of sadow banking s, invesmen risk x q, and profiabiliy of banking R r τ as funcions of e sae variable ω i.e., bankers weal sare in e modified model wi a consan opporuniy cos of defaul H = 2.88, wic equals e average cos of defaul in e calibraed model in Secion 2.3 in BSB. For oer parameer values see e same secion. 3

14 fracion of capial eld ψ.5 by bankers q price of pysical capial.2 s + s 5 5 overall leverage a b c s 2.. leverage of sadow banking d x q invesmen risk e R r τ.3.2. profiabiliy of banking f Figure 3: Tis figure presens e fracion of pysical capial eld by bankers ψ, e price of pysical capial q, banker s overall leverage s + s, e leverage of sadow banking s, invesmen risk x q, and profiabiliy of banking R r τ as funcions of e sae variable ω i.e., bankers weal sare in e modified model in wic ouseolds ave Epsein-Zin preferences. For e coice of parameer values, see Secion 3. 4

15 x q financial insabiliy ax rae,τ 3 leverage of sadow banking s ax rae,τ Figure 4: Tis figure sows ow e cange in e ax rae influences e invesmen risk x q e upper panel and e leverage of sadow banking e lower panel a e socasic seady saes of e modified model wi ouseolds of Epsein-Zin preference. We use e saionary disribuion o calculae momens. For parameer values oer an τ, see Secion 3. 5

16 56 banker s welfare 27 ouseold s welfare ax rae, τ a aggregae welfare ax rae, τ c ax rae, τ b average risk-free rae ax rae, τ d Figure 5: Welfare Tis figure sows ow e represenaive banker s welfare panel a, e represenaive ouseold s welfare panel b, e sum of e wo agens welfare panel c, and e average risk-free rae panel d cange wi e ax rae. For agens welfare, we focus on sae ω =.38. For parameer values oer an τ, see Secion 3. 6

17 average grow rae of e banker s weal ax rae, τ a average grow rae of e ouseold s weal ax rae, τ c average volailiy of e banker s weal ax rae, τ b average volailiy of e ouseold s weal ax rae, τ d Figure 6: Welfare Tis figure sows ow ax rae influences e average grow rae lef panels and e average volailiy rig panels of bo e represenaive banker s weal upper panels and e represenaive ouseold s weal lower panels. We use e saionary disribuion o calculae momens. For parameer values oer an τ, see Secion 3. 7

18 ψ fracion of capial eld.5 by bankers q price of pysical capial s leverage of regular banking.2.4 a b.2.4 c leverage of sadow banking.4 invesmen risk.3 profiabiliy of banking s. 5. x q R r τ d e.2.4 f Figure 7: Tis figure presens e fracion of pysical capial eld by bankers ψ, e price of pysical capial q, e leverage of regular banking s + s, e leverage of sadow banking s, invesmen risk x q, and profiabiliy of banking R r τ as funcions of e sae variable ω i.e., bankers weal sare in e modified model wi e capial requiremen consrain. Te coice of parameer values follows: = 3%, χ = %, a =.225, a =., δ = %, φ = 3, λ =, x = 4%, and s =

19 Financial Insabiliy. x q maximum leverage of regular banking, s. Leverage of Sadow Banking s maximum leverage of regular banking, s Figure 8: Tis figure sows ow e cange in capial-requiremen consrains influences e invesmen risk x q e upper panel and e leverage of sadow banking e lower panel a e socasic seady saes of e modified model. Te socasic seady sae is e sae were ωµ ω λω ˆω =. Te maximum leverage of regular banking s is 2.8. For e coice of oer parameer values, see Figure 7. 9

20 References Campbell, Jon Y and Jon H Cocrane 999 By Force of Habi: A Consumpion Based Explanaion of Aggregae Sock Marke BeaviorV, Journal of Poliical Economy, Vol. 7, pp Crisiano, Lawrence J, Marin Eicenbaum, and Carles L Evans 25 Nominal Rigidiies and e Dynamic Effecs of a Sock o Moneary Policy, Journal of Poliical Economy, Vol. 3, pp. 45. He, Ziguo and Arvind Krisnamury 22 A Macroeconomic Framework for Quanifying Sysemic Risk, Fama-Miller Working Paper, pp Louskina, Elena 2 Te Role of Securiizaion in Bank Liquidiy and Funding Managemen, Journal of Financial Economics, Vol., pp Sokey, Nancy, Rober Lucas, and Edward Presco 989 Recursive Meods in Economic Dynamics, Cambridge MA. Wacer, Jessica A 23 Can Time-Varying Risk of Rare Disasers Explain Aggregae Sock Marke Volailiy? Te Journal of Finance, Vol. 68, pp