Asset Portfolio Choice of Banks and Inflation Dynamics

Transcription

1 Bank of Japan Working Paper Serie Aet Portfolio Choice of Bank and Inflation Dynamic Kouke Aoki * kaoki@e.u-tokyo.ac.jp Nao Sudo ** nao.udou@boj.or.jp No.12-E-5 July 212 Bank of Japan Nihonbahi-Hongokucho, Chuo-ku, Tokyo 13-21, Japan * Aociate Profeor, Faculty of Economic, Univerity of Tokyo ** Deputy Director, Reearch and Statitic Department Paper in the Bank of Japan Working Paper Serie are circulated in order to timulate dicuion and comment. View expreed are thoe of author and do not necearily reflect thoe of the Bank. If you have any comment or quetion on the working paper erie, pleae contact each author. When making a copy or reproduction of the content for commercial purpoe, pleae contact the Public Relation Department (pot.prd8@boj.or.jp) at the Bank in advance to requet permiion. When making a copy or reproduction, the ource, Bank of Japan Working Paper Serie, hould explicitly be credited.

2 Aet Portfolio Choice of Bank and Inflation Dynamic Kouke Aoki and Nao Sudo July 11, 212 Abtract Since the mid-199, the aet portfolio of Japanee bank have continuouly tilted toward government bond, while lending to firm ha declined. In thi paper, we invetigate the caue and conequence of uch change in bank behavior by introducing bank aet portfolio deciion into an otherwie tandard New Keyneian dynamic tochatic general equilibrium model. In our model, bank contruct their portfolio under the value at rik contraint, which require bank to repay their debt regardle of the return on their aet or whether the maximum lo on their aet materialized. A a reult, the maximum lo on aet and bank net worth affect bank balance heet and aet portfolio allocation by changing their rik taking capacity. For intance, an increae in downide rik or a deterioration in bank net worth reduce their rik taking capacity, and reult in a contraction of their balance heet a well a rebalancing of their portfolio toward government bond, thu dampening output and inflation. We etimate the model by Bayeian etimation and find that uch portfolio deciion played an important role in the accumulation of government bond and deflation in Japan ince the latter half of the 199. Keyword: Value at Rik Contraint; Bank Aet Allocation; Deflation; Lot Decade. The author would like to thank Takatohi Ito, Tohiki Jinuhi, Munehia Kauya, Koichiro Kamada, Takehi Kimura, Takahi Kozu, Eiji Maeda, Koji Nakamura, Kenji Nihizaki, Maahi Saito, Takatohi Sekine, Youke Takeda, Tutomu Watanabe, participant of the fourth Univerity of Tokyo Bank of Japan conference in November 211, and the taff of the Reearch and Statitic Department, the Bank of Japan, for their ueful comment. The view expreed in thi paper are thoe of the author and do not necearily reflect the official view of the Bank of Japan. Aociate Profeor, Faculty of Economic, Univerity of Tokyo ( kaoki@e.u-tokyo.ac.jp) Deputy Director, Reearch and Statitic Department, Bank of Japan ( nao.udou@boj.or.jp). 1

3 1 Introduction Japan long-lating economic tagnation ince the beginning of the 199, or what i increaingly coming to be called the two lot decade, i, in general, attributed to advere environmental change in the real ide of the economy, uch a the permanent lowdown of total factor productivity in the early 199, or the introduction of the mandatory reduction of working hour in the late The pioneering tudy by Hayahi and Precott (22), baed on a imple growth model, how that Japan economic downturn in the 199 can be explained by an exogenou decline in total factor productivity growth. Another trand of tudie, uch a Bayoumi (21), Hohi and Kahyap (24, 21), Caballero, Hohi, and Kahyap (28), and Hiroe and Kurozumi (21), by contrat, ugget that malfunction of the banking ector play a large role in explaining Japan two lot decade. Bayoumi (21), for intance, uing vector autoregreion, argue that diruption in financial intermediation due to the deterioration of bank balance heet and the need to meet capital adequacy tandard i the major explanation for the lowdown of the economy. In fact, the banking ector ha been ubject to contant change in the economic environment over the pat two decade. From the beginning of the 199 onward the bank needed to rebuild their balance heet and reduce lending to firm in the face of mounting bad loan originating from the burt of the aet bubble and the full-cale enforcement of Bael capital adequacy requirement. Before the economy, including the banking ector, wa able to embark on a recovery path, Japan uffered a banking crii triggered by the collape of Sanyo Securitie and Yamaichi Securitie, leading to a deteriorationinbank profit tructure and balance heet. 2 A a reult, the diruption of financial intermediation worened further. The aim of thi tudy i to examine the econd explanation of Japan diappointing performance focuing on malfunction in the financial ector further by crutinizing the reaon for, and conequence of, economic activitie in the banking ector. To thi end, we hed light on one other peculiar change in the economic environment during thetwolotdecadeofthe199and2. Thati,wefocuontheecularincreae in government bond iuance, particularly from the latter half of the 199. Figure 1 diplay the time path of government bond outtanding relative to GDP and the aggregate capital tock. 3 The figure clearly how that the amount of government bond outtanding ha grown more quickly than GDP, and the aggregate aet portfolio i 1 Alo ee Otu (211) for the role played by the wedge aociated with the labor upply deciion in explaining the Japanee buine cycle. 2 Kaihatu and Kurozumi (21) develop a model incorporating both friction aociated with the non-financial part of the economy and with the financial ector and quantitatively examine their relative importance in explaining Japanee buine cycle. 3 Unle otherwie noted, government bond include treaury dicount bill, central government ecuritie and FILP bond, local government ecuritie, and public corporation ecuritie. 2

4 tilted toward government bond. 4 Thi acceleration of government debt accumulation i cloely related to bank adjutment of their balance heet and of the compoition of their aet portfolio. Figure 2 how change over the pat three decade in bank aet allocation between government bond purchae and loan to firm, a well a the ratio of the amount of government bond outtanding held by bank over total government debt. A can be clearly een, bank purchae of government bond tarted to rie in the mid-199, while bank loan claim began to decline at around the ame time. Conequently, a the bottom panel in Figure 2 indicate, the bulk of the increae in government bond outtanding during the two lot decade ha been aborbed by the banking ector. Bank aet portfolio allocation thu ha played a key role in the teady demand for government debt a the iuance of government bond increaed over time. How then do the change in bank economic activitie affect fluctuation in the macroeconomy. Figure 3 how the path of GDP growth and inflation over time. GDP growth remained trong up until the beginning of the 199, but declined immediately after the burt of the bubble. Following a moderate recovery in the middle of the 199, GDP growth declined further after the banking crii and never reverted back to the rate een in the 198. Meanwhile, inflation wa poitive during the 198, began to weaken after the burt of the bubble in February 1991, 5 and ha been negative ince November 1997, the tart of the banking crii that gripped Japan for a number of year. 67 In thi paper, we explore the linkage between the macroeconomic variable and bank adjutment of their balance heet and allocation of aet, both from a theoretical and an empirical perpective. To thi end, we develop a model that incorporate bank and government bond into an otherwie tandard New Keyneian dynamic tochatic general equilibrium (DSGE) model. In the model, bank collect depoit from houehold, and invet the depoit and their own net worth in two aet: loan to firm, which are equivalent to invetment in productive capital, and government bond. Bank decide the ize of their balance heet and their aet portfolio allocation between the two aet, o a not to violate the value at rik (VaR) contraint. Under the VaR contraint, bank have to repay all of their debt to houehold, regardle of the ex-pot return on the two type of aet. Since the ex-pot return from holding the aet are uncertain at the timing of the adjutment of aet portfolio, and can be lower than the depoit 4 There i a growing literature on the accumulation of government bond in the Japanee economy from the perpective of government debt utainability, including Doi, Hohi, and Okimoto (211) and Imrohologlu and Sudo (211). See Enomoto and Iwamoto (28) for the welfare implication of fical policy undertaken during the two lot decade. 5 There are everal different view about when the bubble burt. February 1991 i the peak of the economic expanion that tarted from November 1986, dated by the cabinet office. 6 Sugo and Ueda (28), etimating a dynamic tochatic general equilibrium (DSGE) model à la Smet and Wouter (23) baed on Japanee data, report that mot of the variation in long-run inflation i accounted for by variation in the target in the monetary policy rule. 7 Hayakawa and Maeda (2) and Sudo (211) argue that the banking crii hampered financial intermediation, encouraged houehold to engage in precautionary aving, and lowered the velocity of circulation of money and the price level. 3

5 rate, bank contruct their aet portfolio uch that they remain olvent even if the maximum loe on each type of aet i realized. Our VaR contraint i imilar to the VaR contraint analyzed by Adrian and Shin (211). In their tudy, bank invet external fund and their own net worth only in capital good, and the VaR contraint work a a ource of fluctuation in the ize of bank balance heet. By contrat, in our model, there are two aet in which to invet, and the VaR contraint alo work a a ource of compoitional change in bank aet portfolio allocation between government bond and loan to firm. The central purpoe of introducing the VaR contraint i that it allow u to incorporate bank rik taking capacity into the model. 8 When there i no VaR contraint in the economy, bank optimal aet portfolio deciion require that the expected return from the two aet are equalized in equilibrium. However, in the preence of a VaR contraint, bank aet portfolio depend not only on the expected return of the two aet, but alo on the maximum lo on the two type of aet and bank net worth. For intance, when the maximum lo of holding a loan claim increae, bank rearrange the ize of their balance heet and the compoition of aet portfolio o a to avert bankruptcy. Becaue uch downide rik reduce bank rik taking capacity, bank remain olvent in the wort-cae cenario by hrinking their balance heet and inveting more in aet with a maller maximum lo. Change in the intitutional environment, uch a the trengthening of capital requirement, may affect the economy in the imilar manner. 9 Such change in the intitutional environment may lead bank to rearrange their balance heet and aet portfolio by directly affecting their rik taking capacity. Bank net worth alo play a ignificant role in their aet portfolio deciion. When their net worth deteriorate, bank repayment capacity in the wort-cae cenario become maller than would otherwie be the cae. In uch a ituation, bank, a hown by Adrian and Shin (211), avert bankruptcy by reducing their balance heet and reallocating their aet portfolio from aet with a larger maximum lo to aet with a maller maximum lo. Next, we dicu the implication of bank invetment deciion under the VaR contraint for the dynamic of output and inflation. Suppoe that uncertainty regarding the ex-pot capital return rie and hence the maximum lo on loan claim holding increae. In thi cae, bank facing a VaR contraint will reduce their invetment in loan claim, a they adjut the ize of their balance heet and the compoition of their aet portfolio. A a reult, the upply of capital to good producing firm will fall, reducing 8 In the preent paper, we focu on an economy where bank rik taking capacity i limited becaue of the VaR contraint. Conequently, capital invetment by bank i lower when compared with an economy where uch a contraint i abent. By contrat, recent tudie, including Korinek (211) and Kato and Turuga (211), invetigate an economy in which exceive invetment by an individual bank lead to negative externalitie, uch a a fall in aet price driven by the fire-ale of aet. 9 Gerali et al. (29), employing a model in which the interet rate at which bank lend to firm decreae with bank net worth, how that a deterioration in bank net worth or a trengthening of capital requirement may increae the interet rate at which bank lend to firm and hence dampen lending and output. 4

6 output. A output decline, aggregate demand weaken and inflation decline. The initial effect, originating from the change in the maximum lo, bring about econd-round effect on the macroeconomy through endogenou development in the bank net worth. When the initial effect lead to a decline in bank net worth, it alo dampen output and inflation through change in the rik taking capacity originating from inufficiencie in bank net worth. The implication of our model are conitent with Japan experience ince 199. A we aw above, Japanee bank experienced an increae in capital requirement, an accumulation of bad loan triggered by the burt of the aet bubble, and a deterioration in their balance heet temming from the banking crii that tarted in In our model, all of thee event may lead to a hrinkage of bank balance heet and a reallocation of bank aet portfolio toward government bond, generating downward preure on economic activity and inflation. To ee how well our model account for development in Japanee economy, both quantitatively and qualitatively, we make ue of Bayeian technique and etimate the parameter of the model and underlying hock baed on data from 198 to 27. We confirm that the VaR contraint play an important role in explaining macroeconomic fluctuation through it effect on bank behavior. In particular, we find that hock to bank net worth contribute ignificantly to the peritent deflation ince the outbreak of the banking crii in Shock to the maximum lo on loan claim alo play a izable role. In addition, we find that the preence of the VaR contraint increae the volatility of buine cycle through the endogenou development in bank net worth. Under the VaR contraint, the macroeconomic outcome of the exogenou hock are amplified by affecting bank balance heet and aet portfolio allocation. Our analyi i cloely related to Braun and Nakajima (211), which examine the impact of accumulated government debt on the price level, focuing on bank aet allocation. Bank in their model hold government bond a collateral to finance their aet purchae. A long a a certain hare of bank are optimitic about future bond price, thee bank purchae government bond by raiing fund from other agent, uing government bond a collateral. A a reult, an accumulation of government bond and deflation coexit in the economy. Our paper alo highlight the effect of bank aet allocation. The economic mechanim that lead bank to hold government bond, however, differ from the one dicued in Braun and Nakajima (211). In our paper, the key determinant of bank aet portfolio i the everity of the VaR contraint. Whenever the contraint tighten, bank become more inclined to reallocate their portfolio from riky to le riky aet. Another tudy with which the preent paper i related i that by Brunnermeier and Sanikov (211). In their model, there are market imperfection in financial intermediation activity. Whenever the economy i hit by an advere hock, agent reallocate their aet portfolio toward afe aet, reulting in deflation. Becaueafeaetinominal aet, higher demand for uch aet raie their value, leading to a fall in the price 5

7 level. Although a imilar mechanim i preent in our model, how bank aet portfolio allocation affect inflation dynamic differ between the two model. In our model, bank flight to afe aet impede capital accumulation in the economy, and dampen output. Inflation fall becaue aggregate demand decline. Regarding the role played by uncertainty, our analyi i alo related to the work by Fernandez-Villaderde et al. (211). Uing tructural vector autoregreion, they empirically how that higher volatility in productivity lower the price level and output, and provide a theoretical framework for analyzing the relationhip between uncertainty and houehold aet allocation. Further, uing the inventory model of money demand, they how that houehold facing greater uncertainty prefer afer and more liquid aet, uch a money, to rikier aet. The ret of the paper i organized a follow. Section 2 preent our model with bank that endogenouly chooe their aet portfolio under the VaR contraint. In addition, we explore the qualitative propertie of our model uing a implified etting. Section 3 conider the quantitative implication of our model baed on parameter etimated uing Japanee data from 1981Q1 to 27Q4. Section 4 conclude the analyi and dicue future extenion of our analyi. 2 The Model Thi ection decribe the tructure of our model. The economy that we model conit of even type of agent: a repreentative houehold, bank, intermediate good producer, wholeale good producer, final good producer, the government and the central bank. See Figure 4 for a graphic repreentation of the model. The repreentative houehold upplie labor to intermediate good producer, receive wage, make depoit at the bank, and receive the repayment of the depoit in turn. The houehold ha no mean of acceing the financial market directly and cannot own the financial aet other than bank depoit. Bank collect depoit from the houehold, and invet their own net worth and depoit in two type of aet: loan for capital good ued by intermediate good producer and government bond. Bank chooe their aet portfolio allocation uch that they do not violate the VaR contraint. Intermediate good producer hire labor and capital good from the houehold and bank, repectively, to produce final good. Wholeale good producer produce differentiated wholeale good from intermediate good. They are monopolitic upplier of wholeale good, and et theirpriceoatomaximizeprofit. Finalgoodproducerconvertthedifferentiated wholeale good into final good. The government collect a lump-um tax from the houehold and iue government bond to financegovernment debtandgovernment expenditure. The central bank control inflation by adjuting the nominal interet rate, according to a Taylor rule. 6

8 2.1 Houehold The infinitely-lived repreentative houehold make deciion on conumption and depoit holding. It i barred from the financial market and thu poee no real capital tock or government bond, and hold all of it aving in the form of bank depoit. The houehold preference with regard to conumption good c ( t ) and work effort l ( t ) are preented in the following expected utility function, (1) E X t= β t U(c t,l t )=E X t= β t log c t + η log 1 l t, (1) where β (, 1) i the dicount factor and η i the weight aigned to leiure. The budget contraint of the houehold i given by the following equation: c t + d t = r d t 1 d t 1 + W (t ) P ( t ) l t + Π t τ t (2) where d ( t ) repreent the houehold depoit, r d ( t 1 ) i the real depoit rate paid by bank for depoit made in period t 1, W (h, t ) i the nominal wage rate, P ( t ) i the price index, Π ( t ) i the um of intermediate good producer and bank real profit that are returned to the houehold a dividend, and τ ( t ) i the lump-um real tax collected by the government. We aume that depoit are a rik-free aet and the real depoit rate i the real rik-free rate. The firt-order condition aociated with the houehold intertemporal conumption deciion i given by U c (c t,l t )=βr d t E t U c (c t+1,l t+1 ), where U c (c ( t ),l( t )) denote the marginal utility with repect to conumption in period t. Becaue the houehold only financial aet are it bank depoit, it conumption growth depend on the rik-free rate. The firt-order condition aociated with the houehold intra-temporal conumptionleiure deciion i given by U l (c ( t ),l( t )) U c (c ( t ),l( t )) = W (t ) P ( t ), where U l (c ( t ),l( t )) denote the marginal utility with repect to leiure in period t. 2.2 Bank Outline of bank choice There i a continuum of rik-neutral bank, indexed by i (, 1). Each bank i collect depoit d (i, t ) from the houehold and purchae loan claim, i.e., capital tock 7

9 k (i, t ), and real government bond b (i, t ) B(i,t ), from final good producer and the P ( t ) government, repectively. Bank finance thee purchae uing the depoit d (i, t ) they have collected and their own real net worth n (i, t ). Bank i balance heet each period i therefore given by k i, t + B (i, t ) = n i, t + d i, t. (3) P ( t ) Bank i receive return on the two type of aet it inveted in in the previou period, repay the depoit to the houehold, and retain the ret of the earning a part of it own net worth. Conequently, the bank net worth evolve according to the following law of motion n i, t+1 = r k t+1 k i, t + r b t+1 b i, t r d t d t i, t, (4) where r k ( t+1 ) and r b ( t+1 ) are the ex-pot real return on loan claim and government bond, repectively. Note that the real return on government bond i given by the policy rate R B ( t ) et by the central bank, divided by the inflation rate π ( t+1 ), that i: r b t+1 = R B ( t ) π ( t+1 ). Bank i accumulate net worth up until the period when it exit from the economy. 1 We aume that bank i exit probability each period i exogenouly given by 1 γ ( t ). The continuation value of bank i i then given by V n i, t = βe t Λ t,t+1 γ t V n i, t γ t n i, t+1, (5) where n (i, t ) i the net worth of bank i and Λ t,t+1 denote the houehold tochatic dicount factor from period t to period t +1. In chooing it portfolio allocation between the two type of aet, bank i conider a VaR contraint imilar to the one dicued in Adrian and Shin (211), together with the expected average return of the two type of aet. Namely, bank i adjut it balance heet in period t, o that it i able to repay all of it debt to the houehold, even if the two type of aet yield the maximum lo in period t +1. Denoting the maximum lo from holding the two type of aet by E t r k ( t+1 ) and E t r b ( t+1 ), repectively, the VaR contraint i given by E t r k t+1 k i, t + E t r b t+1 b i, t r d t d i, t. (6) 1 Following Gertler and Karadi (211), we aume that a bank tranfer all of it accumulated net worth to the houehold when it exit from the economy. 8

10 Here, we aume that loan claim holding have a larger rik aociated with them than government bond holding, o that E t r k ( t+1 ) < E t r b ( t+1 ). 11 There are two poible interpretation why the maximum lo aociated with a particular type of aet might vary over time. The firt interpretation i that it might reflect hock to the economic environment, uch a an increae in downide rik of the ex-pot return of an aet or an increae in uncertainty regarding the macroeconomic outlook. The econd interpretation i that it might reflect intitutional change, uch a a trengthening of capital requirement. Such intitutional change directly limit bank rik taking capacity, generating imilar conequence to thoe under the firt interpretation. 12 Bank maximization problem In Adrian and Shin (211) model, where there i only one type of aet, the VaR contraint matter only for the ize of a bank leverage. By contrat, in our model with two type aet, the VaR contraint influence the aet portfolio allocation a well a the ize of leverage. Bank i optimization problem i formulated a the maximization of it net worth in the lat period prior to exiting from the economy, which i hown by equation (5), ubject to bank i balance heet equation (3), the law of motion for bank i net wort accumulation (5), and the VaR contraint (3). Becaue bank are rik-neutral, we firt gue that the value function of the bank i i given by Equation (5) then reduce to V n i, t = φ t n i, t. max V n i, t = βe t Λ t,t+1 γ t φ t+1 q k ( t+1 ) k (i, t ) +q b ( t+1 ) b (i, t ) +r d ( t+1 ) n (i, t ) + 1 γ t q k t+1 k i, t + q b t+1 b i, t + r d t n i, t. 11 In the preent paper, we concentrate our analyi on an equilibrium where bank hold both type of riky aet, and the wort-cae return of the two type of aet are maller than the rik-free rate, o that following the two equation hold: r k t+1 t r d t <, r b t+1 t r d t <. 12 Admittedly, the maximum lo on loan claim or government bond may be endogenouly affected by the current economic environment, uch a the amount of capital tock available in the economy or the type of government policy being purued. Here, however, we aume that the law of motion for the maximum lo i given and concentrate our analyi on how variation in the maximum lo affect the economy. 9

11 The correponding firt order condition yield " # " # (γφ( t+1 )+1 γ( t )) Λ t,t+1 q k ( t+1 ) (γφ( t+1 )+1 γ( t )) Λ t,t+1 q b ( t+1 ) E t = E q k ( t+1 t. ) q b ( t+1 ) (7) Here q k ( t+1 ) r k ( t+1 ) r d ( t ) and q b ( t+1 ) r b ( t+1 ) r d ( t ) repectively denote the exce return to loan claim holding and government bond holding relative to depoit, repectively. Similarly, q k ( t+1 ) r k ( t+1 ) r d ( t ) and q b ( t+1 ) r b ( t+1 ) r d ( t ) denote the exce return to the two type of riky aet when the wort poible return i realized. Equation (7) provide the fundamental principle baed on which bank i allocate it aet to loan claim and government bond. When the VaR contraint i effective, it i not neceary that the expected exce return of the two type of aet are equalized in equilibrium. Intead, bank i aet portfolio i contructed o that the expected exce return weighted by the maximum lo for each type of aet are equalized. Under the premie that loan claim are rikier than government bond, i.e., r k ( t+1 ) <r b ( t+1 ), the expected exce return on loan claim need to exceed that on government bond, i.e., E t r k ( t+1 ) > E t r b ( t+1 ), to compenate for thi. From equation (6) and (7), we obtain the expreion for φ ( t ). φ t = βe t hλ t,t+1 γ t φ t+1 +(1 γ t ) ª r ³ d t 1 q i k t+1 /q k t+1. (8) Aggregation Bank exit from the economy with probability 1 γ ( t ) each period, and bank aggregate net worth, i.e., the um of all bank net worth, evolve according to the following law of motion: n t = γ t r k t k t 1 + r b t b t 1 r d t 1 d t 1, where n ( t ) i aggregate bank aggregate net worth. An increae in the exit probability reduce bank net worth. A hown in equation (6), the reduced net worth lead to a tightening of bank VaR contraint, affecting the ize of bank balance heet and the compoition of their aet portfolio in the ubequent period. In addition to the fundamental earning from invetment in loan claim and government bond, the accumulation of aggregate net worth i affected by exogenou hock to the exit probability γ ( t ). Such exogenou hock include phenomena uch a aet bubble, irrational exuberance, or an 1

12 innovation in the efficiency of bank invetment Intermediate Good Producer Intermediate good producer produce intermediate good y ( t ), and ell them to wholeale good producer at price P y ( t ). They hire labor input l ( t ) from the houehold and borrow effective capital v ( t ) K ( t 1 ) from bank. Both the input and output market of intermediate good producer are competitive. The maximization problem for intermediate good producer i given by P y ( t ) y ( t ) max r t v t k t 1 W t l t, y( t ),v( t )k( t 1 ),l( t ) P ( t ) ubject to y t = v t k t 1 α A t Z t l t 1 α, (9) where v ( t ) i the capital utilization rate, k ( t 1 ) i the capital tock, r ( t ) i the real return to the ue of effective capital, A ( t ) i the tationary component of the technology level, Z ( t ) i the non-tationary component of the technology level, and α [, 1] i the capital hare. The firt order condition for intermediate good producer yield the following equalitie. r k t = α P y ( t ) v t k t 1 α 1 A t Z t l t 1 α, P ( t ) W ( t ) = P y ( t ) P ( t ) P ( t ) (1 α) v t k t 1 α A t Z t 1 α l t α. The capital utilization rate i determined by the bank. Auming that chooing capital utilization v ( t ), together with the capital tock k ( t 1 ), bank incur the real cot of κ v k ( t 1 )(v( t )) φ+1 1, φ Baedonthefinancial accelerator model developed by Bernanke, Gertler, and Gilchrit (1999), Gilchrit and Leahy (22) and Nolan and Thoenien (29) examine the conequence of an exogenou deterioration in entrepreneurial net worth, which i imilar to the exogenou change in γ ( t ) in our model. 14 There are alternative way to incorporate exogenou hock to bank net worth into the model. In Gertler and Karadi (211) model, for example, the exiting capital tock become out of date, reulting in a deterioration in the value of bank loan claim and net worth. On the other hand, in Aoki and Nikolov (211) tudy, which analyze the conequence of bank invetment on the unproductive bubble, the collape of the bubble lead to a deterioration in bank net worth. 11

13 o that bank optimal capital utilization rate i expreed by r k t =(φ +1)κ v v φ t, where κ v and φ are parameter that govern the capital utilization rate. Conequently, bank net return to invetment on productive capital r k ( t ) i given by r k t k t = r k t k t κ v k t 1 v t φ+1 +(1 δ) k t, where δ [, 1] i the depreciation rate of the capital tock. Similarly, the real wage paid to the houehold i expreed by W ( t ) P ( t ) = t (1 α) v t k t 1 α A t 1 α Z t 1 α l t α. 2.4 Wholeale and Final Good Producer Optimization problem of wholeale and final good producer The wholeale good ector contain a continuum of firm, each producing differentiated product, a indexed by z [, 1], from intermediate good employing a linear production technology; x(z, t )=y(z, t ). Here, x(z, t ) denote the differentiated wholeale good made by wholeale good producer z and y(z, t ) itheintermediategooduedainputbyproducerz. Final good producer purchae thee differentiated good in a competitive market, producing the final good from wholeale good employing the following contant elaticity of ubtitution (CES) aggregate technology: x " Z # t 1 ε( ε ( t ) t ) 1 1 ε( t ) = x( t ε(,z) t ) dz, ε t > 1 where ε ( t ) (1, ) denote the time-varying elaticity of ubtitution between differentiated wholeale good. Given thi CES technology for final good, the demand for each differentiated wholeale good x (z, t ) i given by a function of it price p(z, t ), the aggregate price index P ( t ), and the aggregate demand for final good x ( t ): µ p(z, x(z, t t ε( ) t ) )= x t. P ( t ) Each wholeale good producer z maximize it profit by chooing the optimal product price. The maximization problem of each differentiated wholeale good producer i given by 12

14 max E t p(z, t+j ) X j= β j Λ j 1,j ³ p(z, t+j ) 1 ε(t ) P ( t+j ) x ( t+j ) ³ ³ Py( t+j ) p(z, t+j ) ε(t ) P ( t+j ) P ( t+j ) x ( t+j ) µ κ p(z, t+j ) p(t+j 1 ) 2 p(z, t+j 1 ) p( t+j 2 ) 2 ³ p(z, t+j ) P ( t+j ) ε(t ) x ( t+j ), where the third term denote the adjutment cot that the wholeale good producer pay for changing it product price p(z, t ), and κ i the parameter that govern the ize of the adjutment cot. Becaue in the ymmetric equilibrium all differentiated good price p(z, t ) et by wholeale good producer are identical, we can obtain the Phillip curve of the economy from the firt order condition of the firm maximization problem ε t µ 1 P y ( t ) P ( t ).5 π t κ π t 1 π t + βκ π t+1 1 π t+1 x ( t+1 ) x ( t ) =. (1) by Market clearing condition The market clearing condition for intermediate good and wholeale good are given Z 1 x t,z dz = y t, x t = Z 1 x t,z dz Final good erve for houehold conumption, invetment in productive capital, and government expenditure. The market clearing condition for final good i given by c t +k t (1 δ) k t 1 +G t = x t κ The government and the central bank π t 1 2 x t κ v k t 1 v t φ+1 The government collect a lump-um tax P ( t ) τ ( t ) from the houehold and iue government bond B ( t ) to finance it repayment R B ( t 1 ) B ( t 1 ) to bank a well a 13

15 government expenditure P ( t ) G ( t ). We aume that a balanced budget i maintained in each period t, thati: R B t 1 B t 1 + P t G t = P t τ t + B t. (11) The government lump-um tax i an increaing function of the amount of outtanding government bond and i pecified a follow: τ t = b t 1 µ b ( t 1 ψ ) T, (12) x ( t ) where ψ (1, ] i the elaticity of the lump-um tax with repect to the amount of outtanding government debt, with an increae in debt leading to an increae in the lump-um tax and T i a contant parameter. Next, we turn to the central bank. The central bank et the nominal interet rate according to a imple Taylor rule given by ln R B t =(1 ρ M )lnr + ρ M ln R B t 1 +(1 ρ M ) φ ln π t + ² r t, (13) where R i contant, ρ M [, 1] i the autoregreive coefficient of the policy rate, and φ > 1 i the policy weight attached to the inflation rate and ² r ( t ) i an i.i.d. hock to the monetary policy rule Shock Proce The exogenou hock in our economy, a hock to the markup-related elaticity ε ( t ), to bank net worth γ ( t ), to the maximum lo on capital aet r k ( t ), to the maximum lo on government bond r b ( t ), to the tationary component of technology A ( t ), to the non-tationary component of technology Z ( t ), and to government expenditure 15 In our model, the policy parameter ψ and φ are both greater than unity, implying that our economy i in the Ricardian regime with regard to both fical and monetary policy. A matter that we do not conider in the preent paper i the cae of government default. In non-ricardian regime with a government default, the inflation rate i only uniquely pinned down when the central bank repond to inflation aggreively. See, for example, Kocherlakota (212). 14

16 G ( t ), evolve according to the equation below: ln ε t = lnε + ² ε t, (14) ln γ t = (1 ρ γ )lnγ + ρ γ ln γ t 1 + ² γ t, (15) ln r k t = (1 ρ rk )lnr k + ρ rk ln r k t 1 + ² rk t, (16) ln r b t = (1 ρ rb )lnr b + ρ rb ln r b t 1 + ² rb t, (17) ln A t = (1 ρ A )lna + ρ A ln A t 1 + ² A t, (18) ln Z t = lnz t 1 + u Z t, (19) u Z t = ρ Z u Z t 1 + ² Z t, (2) ln G t = (1 ρ G )lng + ρ G ln G t 1 + ² G t, (21) where ρ ε, ρ γ, ρ rk, ρ rb, ρ A, ρ Z, and ρ G (, 1) are the autoregreive root of the correponding hock, and ² ε ( t ),² γ ( t ),² rk ( t ),² rb ( t ),² A ( t ),² Z ( t ) and ² G ( t ) are the exogenou i.i.d. hock that are normally ditributed with mean zero. 2.7 Equilibrium Condition An equilibrium conit of a et of price, {W ( t ),P( t ),P y ( t ),r k ( t ), r k ( t ),r d ( t ), r b ( t ),R B ( t )} t=, and the allocation {c ( t ),l( t ),d( t ), Π ( t ),k( t ),v( t ),x( t ), y ( t )}} t=, for a given government policy {G ( t ), τ ( t )} t=, realization of exogenou variable {² ε ( t ),² γ ( t ),² rk ( t ),² rb ( t ),² A ( t ),² Z ( t ),² G ( t ),² r ( t )} t=, expected wort-cae return {r k ( t ),r b ( t )} t=, and initial condition {B 1 }, {d 1 }, {k 1 } uch that for all t, i, and z : (i) the houehold maximize it utility given price; (ii) bank maximize their profit given price and expected wort-cae return; (iii) intermediate good producer maximize their profit given price; (iv) wholeale good producer maximize their profit given price; (v) final good producer maximize their profit given price; (vi) the government budget contraint hold; (vii) thecentralbanketthepolicyratefollowingthetaylorrule;and (viii) market clear. 2.8 Steady State Analyi Before invetigating the dynamic of the model, we explore it mechanim in the teady tate to how the determinant of bank balance heet and aet portfolio allocation. In particular, we focu on how the expected return from holding the two type of riky aet in the teady tate, which we denote by r b and r k, are affected by bank VaR contraint, and how bank deciion with regard to their portfolio allocation between 15

17 government bond b andloanclaimk i made. 16 For illutrative purpoe, we make two implifying aumption in thi ubection: (1) the houehold upplie labor inelatically, i.e., l =1, and (2) bank capital utilization cot i zero, i.e., φ =. 17 Evaluating the variable appearing in the portfolio choice equation, the VaR contraint equation, and the law of motion of bank net worth at the teady tate value, we have r k r d = r b r d, (22) r d r k r d r b (r k r d ) k +(r b r d ) b = r d n, (23) γ n = [(r k r d ) k +(r b r d ) b]. (24) 1 γr d Note that the houehold Euler equation in the teady tate implie that r d = 1 β. The three equation above yield the exce return from holding the two type of riky aet, and the pread of the two type of riky aet: r b r d = 1 γr d (r d r γr b ), d (25) r k r d = 1 γr d (r d r γr k ). d (26) r k r b = 1 γr d (r γr b r k ). (27) d Accordingtoequation(25) and (26), the exce return from holding the two type of riky aet and the pread between them are expreed by the expected maximum lo from holding the two type of riky aet r b and r k, together with bank urvival probability γ. For intance, when the maximum lo of holding loan claim r k rie, which implie the value of r k fall, bank VaR contraint become tighter, reducing bank rik taking 16 The definition of the teady tate in our economy need to be carefully tated. Suppoe that we define the teady tate a a ituation in which there are no exogenou hock and every endogenou variable grow at a contant rate. Bank aet allocation then become indeterminate becaue their portfolio choice depend on the rikine of the aet. In the preent analyi, we define the teady tate following Devereux and Sutherland (21, 211) approach, in which bank take the poibility that the wort-cae cenario with regard to aet return i realized into conideration. Conequently, the rik of holding an aet affect bank portfolio in the teady tate. 17 Thi aumption implie that capacity utilization i unity. 16

18 capacity. If bank maintain the ame amount of loan claim holding, they require a higher pread vi-à-vi government bond to hold loan claim than would otherwie be the cae. The government bond yield i unaffected by the change in r k. The imilar mechanim work if the maximum lo of holding the government bond r b increae. By contrat, a reduction in urvival probability γ lead to a rie in the exce return aociated with the two type of aet. A indicated by equation (24), the maller urvival probability prevent bank from increaing their net worth. Becaue a low level of net worth tighten the VaR contraint by increaing the rik of default, bank hrink their balance heet. Since bank demand for both type of aet fall, the exce return on them need to rie to clear the aet market. Next, we dicu how bank allocate their aet between loan claim k and the government bond b. Becaue the return from holding loan claim r k equal the return to the capital tock in the economy, we have r k = αazk α 1 +(1 δ). Similarly, becaue in equilibrium the amount of loan claim i equivalent to the total capital tock, we have 1 rk (1 δ) α 1 k =. (28) αaz Taking into the conideration that α 1 <, a higher return on loan claim implie a maller amount of loan claim and thu a lower amount of invetment in the economy. Baed on the conideration above, an increae in the maximum lo on loan claim holding or a decline in bank urvival probability reduce their loan to firm through a rie in the return on the capital tock, r k. Bank deciion regarding the holding of government bond i affected by the government policy regarding taxe and budget balance. From equation (11) and (12), we have r b b = Tb ψ + b, rb 1 b = T 1 ψ x = rb 1 T 1 ψ AZk α. (29) Here,we aume that the inflation rate i unity in the teady tate. Thee equation ugget that, for a ψ > 1, bank tilt toward holding government bond a the return on them increae. Under the tax policy decribed by the equation above, an increae in government interet rate payment i met by an equivalent increae in government bond iuance, leading to higher government bond holding by bank. Similar to the mechanim that determine the amount of loan claim k, an increae in the maximum lo of holding government bond, expreed by a decline in r b, or a deterioration in 17

19 bank net worth, expreed by a decline in γ, leadtoanincreaeinbank government bond holding through a rie in government bond yield. Latly, we dicu how bank allocate their aet between government bond holding and loan to firm. From equation (28) and (29), the ratio of government bond holding relative to loan claim i given by b k = rb 1 T 1 ψ r k (1 δ) According to the above equation, any change in the economic environment that increae the return on the two type of riky aet, r b and r k, including an increae in the maximum lo on loan to firm or of holding government bond, or a deterioration of bank net worth, lead bank to purchae more government bond relative to loan claim. 2.9 VaR Contraint and Inflation Dynamic Log-linearizing equation (7) and (13) around the non-tochatic teady tate, we obtain E t µ = rk ˆr k t+1 r k r d µ rd r k r d r d r b r d + µ rk + E t ˆr r d r k t+1 k r d r d r k r d r d r b α E t µ ˆr d t.. rb ˆr b t+1 r b r d µ E t rb ˆr r d r b, t+1 b (3) ˆR b t = φˆπ t. (31) Here ˆλ ( t ) denote the log deviation of variable λ ( t ) from it non-tochatic teady tate value. Taking the following relationhip ˆr b t+1 = ˆR b t ˆπ t+1 into conideration, we have ˆπ t = φ 1 E t ˆπ t+1 + a 1ˆr k t+1 + a 2ˆr k t+1 a 3ˆr b t+1 + a 4ˆr d t. (32) Here, a 1,a 2,a 3, and a 4 are all poitive value that are denoted by a 1 = r b r d r d a 4 = r b r d r b r k r k, a 2 = r b r d, a 3 = r b r d r k r d r b r d r k r d r d r µ b rd r µ d rd + r d. r b r d r k r d r d r b r d r k 18 r b

20 Equation (32) how the qualitative linkage between bank aet allocation and the inflation rate in the economy. Other thing, including inflation expectation E t [ˆπ ( t+1 )], being equal, current inflationideterminedbythefourvariablee tˆr k ( t+1 ), E tˆr k ( t+1 ), E tˆr b ( t+1 ), and ˆr d ( t ). Suppoe the expected return from holding loan claim, E tˆr k ( t+1 ), increae, bank invet more in loan claim and a larger amount of capital i accumulated in the economy, trengthening the economy and generating inflation. A decreae in the maximum lo of holding loan claim E tˆr k ( t+1 ) or an increae in the maximum lo of holding government bond E tˆr b ( t+1 ) caue inflation through a imilar mechanim. Similarly, a rie in the depoit rate ˆr d ( t ) would alo generate inflationary preure in the economy. While a higher depoit rate reduce bank retained earning and hamper their accumulation of net worth, it alo encourage them to invet more in loan claim than government bond, increaing the capital upply in the economy. The underlying mechanim i that, under the premie that Er k < Er b <r d < Er b < Er k, the unconditional expected exce return on government bond i more enitive to a change in the depoit rate than that on loan claim. Conequently, bank aet allocation tilt toward loan to firm, leading to inflation in the economy. 3 Quantitative Analyi In thi ection, we invetigate the quantitative implication of our model, paying attention to the role played by bank VaR contraint. Baed on data for Japan, we firt etimate the model parameter and eight tructural hock: a markup hock, ² ε ( t ); ahockto bank net worth, ² γ ( t ); a hock to the maximum lo on loan claim, ² rk ( t ); ahock to the maximum lo on government bond, ² rb ( t ); a temporary technology hock, ² A ( t ); a permanent technology hock, ² Z ( t ); amonetarypolicyhock,² r ( t ); and a government expenditure hock, ² G ( t ), uing Bayeian technique. We then explore the model equilibrium repone to thee exogenou hock. In particular, we examine how the VaR contraint affect the dynamic of the model in repone to thee hock. Latly, we evaluate the quantitative contribution of each hock in explaining variation in macroeconomic variable oberved in Japan during the pat three decade. 3.1 Data Our benchmark dataet include eight time erie for the Japanee economy from 198Q1 to 27Q4: (1) labor input 18, which correpond to l ( t ) in the model; (2) real private invetment, which i taken from the National Account and correpond to k ( t ) 18 To contruct the labor input erie, we follow the methodology adopted in Hayahi and Precott (22). 19

21 (1 δ) k ( t 1 ) in the model; (3) the um of treaury dicount bill, central government ecuritie and FILP bond, local government ecuritie, and public corporation ecuritie held by dometically licened bank, deflated by the GDP deflator; the erie i contructed from the Flow of Fund Account and the National Account and correpond to b ( t ) in the model; (4) the tock price index of bank, deflated by the GDP deflator, which i contructed from Tokyo Stock Exchange data and the National Account and correpond to n ( t ) in the model; (5)the capacity utilization rate of manufacturing indutry, which i taken from the Indice of Indutrial Production and correpond to v ( t ) in the model; (6)the GDP deflator, taken from the National Account, which correpond to P ( t ) in the model; (7) the call rate et by the Bank of Japan, which correpond to R B ( t ) in the model; and (8) GDP, taken from the National Account, which correpond to x ( t ) in the model. All of the erie, other than erie (5) and (7), are firt differenced. Serie (5) and (7) are ued in level. 3.2 Prior and Poterior Ditribution of Parameter The parameter value ued for our quantitative analyi are reported in Table 1. The parameter value are quarterly unle tated otherwie. Since our model i a tandard New Keyneian model except that it incorporate bank aet portfolio choice, we et ome of the parameter, that are not related to bank portfolio deciion to the conventional value. Thee parameter are reported in Table 1(b). Other parameter are etimated uing Bayeian technique, becaue they are pecific to the current model. The third to fifthcolumnoftable1(a)reportthepriorditribution of the etimated parameter, while the lat three column diplay the poterior mean and the confidence interval of the model parameter. 3.3 Impule Repone In thi ubection, we invetigate the economy dynamic repone to tructural hock. Figure 5 to 12 diplay the economy impule repone function to a negative hock to the technology growth rate, ² Z ( t ); a negative hock to the tationary component of technology, ² A ( t ); a negative hock to bank urvival probability (a deterioration in bank net worth), ² γ ( t ); a negative hock to the maximum lo on loan claim, ² rk ( t ); a negative hock to the maximum lo on government bond, ² rb ( t ); a poitive hock to themonetarypolicyrule,² r ( t ); a poitive hock to the markup, ² ε ( t );andanegative hock to government expenditure, ² G ( t ), repectively. All of the equilibrium path after the hock are approximated by log-linearization around the non-tochatic teady tate. In order to highlight the quantitative role played by the VaR contraint in the economic dynamic, we plot the equilibrium repone to a hock in an alternative economy in which the VaR contraint i abent (labeled No VaR and denoted by the line with black circle) in addition to the equilibrium repone to a hock in an economy where 2

22 the VaR contraint i preent (labeled and denoted by the line with white circle). The etting of the economy with No VaR are equivalent to thoe of our economy, except that there i no contraint equation (6). In thi cae, becaue bank no longer conider the maximum lo on aet and their own net worth in their deciion making, bank leverage and aet portfolio allocation are both independent from thee factor. Conequently, bank invet more in thoe aet that offer a higher expected return, and in equilibrium, the expected return on the two type of aet are equalized. A hown in Figure 5, a permanent downward hift in technology dampen longrun economic activity, including invetment and bank net worth accumulation, and reult in downward preure on current inflation. It i noteworthy that thi hock ha twooppoingeffect on inflation dynamic. Firt, it lower the productivity of good production, which ha a poitive effect on inflation. Second, it reduce the economy output and weaken the houehold demand permanently, which ha a negative effect on inflation. In our etting, the latter effect dominate the former. Compared with the economy in which the VaR contraint i abent, in the benchmark economy the hock generate quantitatively larger macroeconomic effect becaue of the endogenou development in bank net worth. When the technology lowdown reduce bank net worth, bank hrink their balance heet o a to avert default. Conequently, le capital i accumulated in the economy, leading to a further decline in output. Next, Figure 6 how the economic repone when a negative temporary technology hock hit the economy. Thi hock lead to a hort-run economic downturn and a deterioration in bank net worth. A for price dynamic, ince the firt channel dicued above dominate the econd channel, the temporary technology hock lead to higher marginal cot and inflation. In the economy with the VaR contraint, ince net worth repond to the hock cyclically, the downturn in output, the increae in inflation, and the extent of balance heet adjutment all become larger than in the economy without the VaR contraint. Turning to Figure 7, thi depict the macroeconomic conequence of a deterioration in bank net worth. Note that in the economy where the VaR contraint i abent, bank net worth cannot be a ource of economic fluctuation, a bank invetment deciion are unaffectedbytheirnetworth. IntheVaRmodel,thedeteriorationinnetworthbring about a receion and deflation in the economy. There are two ditinct channel through which net worth affect bank invetment deciion. The firt channel i the balance heet effect. A pointed out by Adrian and Shin (211), with a maller net worth, bank hrink their balance heet, reducing the total amount of invetment. Conequently, other thing being equal, both loan claim and government bond purchae fall. The econd channel i the aet allocation effect. When their net worth i low, bank reduce lending more than they reduce government bond purchae, ince the maximum lo on loan claim i larger than that on government bond. Bank aet allocation then tilt toward relatively afer aet and their loe in the wort-cae cenario become maller. 21