Intermediary Leverage Cycles and Financial Stability Tobias Adrian and Nina Boyarchenko

Transcription

1 Intermediary Leverage Cycles and Financial Stability Tobias Adrian and Nina Boyarchenko The views presented here are the authors and are not representative of the views of the Federal Reserve Bank of New York or of the Federal Reserve System

2 Introduction Outline Introduction The Model Solution Distortions and Amplification Extensions T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

3 Introduction Questions about Financial Stability Policy Systemic distress of financial intermediaries raises questions about financial stability policies: How does capital regulation affect the trade-off between the pricing of credit and the amount of systemic risk? How does macroprudential policy function in equilibrium? What are the welfare implications of capital regulation? We develop a theoretical framework to address these questions T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

4 Introduction Our Approach We use a standard macro model with a financial sector and add one key assumption: Funding constraints of financial intermediaries are risk based, so intermediaries have to hold more capital when the riskiness of their assets increases This assumption is empirically motivated from risk management practices and regulatory constraints Equilibrium dynamics capture stylized facts: Procyclical leverage of intermediary balance sheets Procyclical share of intermediated credit Relationship between asset risk premia and intermediary leverage T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

5 Introduction Systemic Risk Systemic risk return trade-off Lower probability of distress corresponds to higher prices of risk Tightening capital requirements decreases probability of distress The relationship between household and capital requirements is inversely u-shaped Volatility paradox Lower contemporaneous volatility is associated with higher probability of distress Lower volatility decreases effective risk aversion of intermediaries, leading to increased leverage and thus increased vulnerability to shocks T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

6 The Model Outline Introduction The Model Solution Distortions and Amplification Extensions T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

7 The Model Economy Structure A t k ht Producers random dividend stream, A t,perunit of project financed by direct borrowing from intermediaries and households i t A t k t Intermediaries financed by households against capital investments C bt b ht Households solve portfolio choice problem between holding intermediary debt, physical capital and risk-free borrowing/lending T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

8 The Model Production Aggregate amount of capital K t evolves as Total output evolves as dk t = (I t λ k )K t dt Y t = A t K t Stochastic productivity of capital {A t = e at } t 0 da t = ādt + σ a dz at p kt A t denotes the price of one unit of capital in terms of the consumption good T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

9 The Model Households Household preferences are: E [ + 0 ] e (ξt+ρht) log c t dt Liquidity preference shocks (as in Allen and Gale (1994) and Diamond and Dybvig (1983)) are exp ( ξ t ) dξ t = σ ξ ρ ξ,a dz at + σ ξ 1 ρ 2 ξ,a dz ξt Households do not have access to the investment technology dk ht = λ k k ht dt T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

10 The Model Households Optimization subject to [ + ] max E e (ξt+ρht) log c t dt {c t,k ht,b ht } 0 dw ht = r ft w ht dt + p kt A t k ht (dr kt r ft dt) + p bt A t b ht (dr bt r ft dt) c t dt and no-shorting constraints k ht 0 b ht 0 T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

11 The Model Intermediaries Financial intermediaries create new capital dk t = (Φ(i t ) λ k ) k t dt Investment carries quadratic adjustment costs (Brunnermeier and Sannikov (2012)) ( ) Φ (i t ) = φ φ1 i t 1 Intermediaries finance investment projects through inside equity and outside risky debt giving the budget constraint p kt A t k t = p bt A t b t + w t T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

12 The Model Intermediaries Risk Based Capital Constraint Risk based capital constraint (Danielsson, Shin, and Zigrand (2011)) 1 α dt k td (p kt A t ) 2 = w t Implies a time-varying leverage constraint θ t = p kta t k t w t = α 1 dt 1 2 d(pkt A t) p kt A t Note that the constraint is such that intermediary equity is proportional to the Value-at-Risk of total assets This will imply that default probabilities vary over time Microfoundation of the risk based capital constraint in a static setting is provided by Adrian and Shin (2010) T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

13 The Model Risk-based Capital Constraints VaR is the potential loss in value of inventory positions due to adverse market movements over a defined time horizon with a specified confidence level. We typically employ a one-day time horizon with a 95% confidence level. Source: Goldman Sachs 2011 Annual Report More T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

14 The Model Commercial Bank Tightening Standards 60 ρ= VIX Credit Tightening Q2 91 Q2 93 Q2 95 Q2 97 Q2 99 Q2 01 Q2 03 Q2 05 Q2 07 Q2 09 Q T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

15 The Model Systemic Distress Solvency risk defined by Term structure of systemic distress τ D = inf t 0 {w t ωp kt A t K t } δ t (T ) = P (τ D T (w t, θ t )) In distress Management changes Intermediary leverage reduced to θ 1 by defaulting on debt Intermediary instantaneously restarts with wealth w τ + D = θ τ D θ w τ D T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

16 The Model Intermediaries Optimization Intermediaries are myopic and maximize a mean-variance objective of instantaneous wealth [ ] dwt γ [ ] 2 V dwt t, max θ t,i t E t subject to the dynamic intermediary budget constraint w t dw t = k t p kt A t (dr kt + (Φ (i t ) i t /p kt ) dt) b t p bt A t dr bt and the risk based capital constraint 1 α dt k td (p kt A t ) 2 = w t w t T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

17 The Model Equilibrium An equilibrium in this economy is a set of price processes {p kt, p bt, C bt } t 0, a set of household decisions {k ht, b ht, c t } t 0, and a set of intermediary decisions {k t, β t, i t, θ t } t 0 such that: 1 Taking the price processes and the intermediary decisions as given, the household s choices solve the household s optimization problem, subject to the household budget constraint. 2 Taking the price processes and the household decisions as given, the intermediary s choices solve the intermediary optimization problem, subject to the intermediary wealth evolution and the risk based capital constraint. 3 The capital market clears: 4 The risky bond market clears: 5 The risk-free debt market clears: 6 The goods market clears: K t = k t + k ht. b t = b ht. w ht = p kt A t k ht + p bt A t b ht. c t = A t (K t i t k t ). T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

18 The Model Related Literature Leverage Cycles: Geanakoplos (2003), Fostel and Geanakoplos (2008), Brunnermeier and Pedersen (2009) Amplification in Macroeconomy: Bernanke and Gertler (1989), Kiyotaki and Moore (1997) Financial Intermediaries and the Macroeconomy: Gertler and Kiyotaki (2012), Gertler, Kiyotaki, and Queralto (2011), He and Krishnamurthy (2012, 2013), Brunnermeier and Sannikov (2011, 2012) T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

19 Solution Outline Introduction The Model Solution Distortions and Amplification Extensions T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

20 Solution Solution Strategy Equilibrium is characterized by two state variables, leverage θ t and relative intermediary net worth ω t ω t = Represent state dynamics as w t w t + w ht = w t p kt A t K t dω t ω t dθ t θ t = µ ωt dt + σ ωa,t dz at + σ ωξ,t dz ξt = µ θt dt + σ θa,t dz at + σ θξ,t dz ξt Risk-based capital constraint implies α 2 θ 2 t = σ 2 ka,t + σ2 kξ,t T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

21 Solution Volatility Risk Leverage Growth Lagged Volatility Growth y = x R 2 = Leverage Growth Lagged VIX Growth y = x R 2 = Data Mean 5% Median 95% β β R Simulation details T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

22 Solution Intermediary Balance Sheets I Intermediated Credit Growth y = x R 2 = Total Credit Growth Leverage Growth Leverage Growth Equity Growth Debt Growth Intermediated Credit Growth Leverage Growth Leverage Growth Total Credit Growth Equity Growth y = x R 2 = Debt Growth T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

23 Solution Intermediary Balance Sheets II Table : Procyclicality of Intermediated Credit Data Mean 5% Median 95% β β R T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

24 Solution Excess Returns Equity Excess Return Leverage Growth y = x R 2 = Financial Sector Equity Return Lagged Leverage Growth y = x R 2 = 0.17 Data Mean 5% Median 95% β β R T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

25 Solution Equilibrium Prices of Risk I Shocks dŷ t = σa 1 (d log Y t E t [d log Y t ]) = dz at d ˆθ t = ( ( [ ]) σθa,t 2 + ) 1 σ2 2 dθt dθt θξ,t E t θ t θ t = σ θa,t σ 2 θa,t + σ2 θξ,t dz at + σ θξ,t σ 2 θa,t + σ2 θξ,t dz ξt. T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

26 Solution Equilibrium Prices of Risk II Price of risk of leverage η θt = (σ ka,t σ a ) σ 2 kξ,t ( 2θ ) tω t p kt β (1 ω t ) σ kξ,t + σ ξ 1 ρ 2 ξ,a Price of risk of leverage is always positive (Adrian, Etula, and Muir (2013)), and depends on leverage growth in a nonmonotonic fashion (Adrian, Moench, and Shin (2010) find a negative relationship) T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

27 Solution Equilibrium Prices of Risk III Realized Mean Return S1B5 S3B5 Mom10 S1B4 S2B3 S2B4 S3B4 S4B5 S1B3 S4B4 S1B2 S3B3 S3B2 Mom S4B3 S2B2 9 Mom 8 S5B5 S4B1 S5B2 S4B2 S5B4 S5B3 S5B1 S3B1 Mom Mom Mom 3 Mom 4 76 S2B1 Mom 5 Mom y 2 3yr 3 4yr 4 5yr 1 2yr S1B1 0 1yr S2B5 2 4 Mom Predicted Expected Return Figure : Source: Adrian, Etula, and Muir (2013) T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

28 Solution Equilibrium Prices of Risk IV Price of risk of output ( η yt = σ a + σ ξ ρ ξ,a σ ) ka,t σ a 1 ρ 2 σ ξ,a kξ,t Switches sign, consistent with insignificant estimates of price of output risk Usually becomes negative when exposure to liquidity shock is small T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

29 Distortions and Amplification Outline Introduction The Model Solution Distortions and Amplification Extensions T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

30 Distortions and Amplification Term Structure of Systemic Risk Distress probability α=2 α= Horizon T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

31 Distortions and Amplification Volatility Paradox Distress probability Distress probability Local volatility Price of leverage risk T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

32 Distortions and Amplification Constant Leverage Benchmark Constant expected output and consumption growth But lower level of output and consumption growth Constant investment and price of capital Liquidity shocks have no impact on real activity T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

33 Distortions and Amplification A Sample Path of the Economy Wealth Equity Return Output Leverage Year Year Year Year 8 6 Consumption T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

34 Distortions and Amplification Household Welfare 1 Welfare Distress probability month 1 year 5 year α α T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

35 Extensions Outline Introduction The Model Solution Distortions and Amplification Extensions T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

36 Extensions Alternative Specification Two financial sectors: banks and funds Leveraged intermediaries have VaR constraint (as in the current paper) while funds have skin in the game constraint (as in He and Krishnamurthy (2012, 2013)) Bank managers, fund managers, and households have log utility VaR constraint sometimes binds T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

37 Extensions π bt w ht C bt b ht Households Invest in risk-free debt, non-bank financial sector and bank financial sector π ft w ht dr f t π ft w ht Banks Create new capital; financed by debt issuance to the households Funds Hold existing capital; financed by profit sharing contracts with households A t k t Φ (i t ) k t Producers productivity A t per unit of project; financed by financial sector At k ht More T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

38 Extensions Additional Research Tradeoff between capital and liquidity regulation Stress tests Intermediation chains More T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

39 Extensions Conclusion Dynamic, general equilibrium theory of financial intermediaries leverage cycle with endogenous amplification and endogenous systemic risk Conceptual basis for policies towards financial stability Systemic risk return trade-off: tighter intermediary capital requirements tend to shift the term structure of systemic downward, at the cost of increased prices of risk today Model captures important stylized facts: 1 Procyclical intermediary leverage 2 Procyclicality of intermediated credit 3 Financial sector equity return and intermediary leverage growth 4 Exposure to intermediary leverage shocks as pricing factor T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

40 Extensions Tobias Adrian and Hyun Song Shin. Procyclical Leverage and Value-at-Risk. Federal Reserve Bank of New York Staff Reports No. 338, Tobias Adrian, Emanuel Moench, and Hyun Song Shin. Financial Intermediation, Asset Prices, and Macroeconomic Dynamics. Federal Reserve Bank of New York Staff Reports No. 442, Tobias Adrian, Erkko Etula, and Tyler Muir. Financial Intermediaries and the Cross-Section of Asset Returns. Journal of Finance, Forthcoming. Franklin Allen and Douglas Gale. Limited market participation and volatility of asset prices. American Economic Review, 84: , Ben Bernanke and Mark Gertler. Agency Costs, Net Worth, and Business Fluctuations. American Economic Review, 79(1):14 31, Markus K. Brunnermeier and Lasse Heje Pedersen. Market Liquidity and Funding Liquidity. Review of Financial Studies, 22(6): , Markus K. Brunnermeier and Yuliy Sannikov. The I Theory of Money. Unpublished working paper, Princeton University, Markus K. Brunnermeier and Yuliy Sannikov. A Macroeconomic Model with a Financial Sector. Unpublished working paper, Princeton University, T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

41 Extensions Jon Danielsson, Hyun Song Shin, and Jean-Pierre Zigrand. Balance sheet capacity and endogenous risk. Working Paper, Douglas W. Diamond and Philip H. Dybvig. Bank runs, deposit insurance and liquidity. Journal of Political Economy, 93(1): , Ana Fostel and John Geanakoplos. Leverage Cycles and the Anxious Economy. American Economic Review, 98(4): , John Geanakoplos. Liquidity, Default, and Crashes: Endogenous Contracts in General Equilibrium. In M. Dewatripont, L.P. Hansen, and S.J. Turnovsky, editors, Advances in Economics and Econometrics II, pages Econometric Society, Mark Gertler and Nobuhiro Kiyotaki. Banking, Liquidity, and Bank Runs in an Infinite Horizon Economy. Unpublished working papers, Princeton University, Mark Gertler, Nobuhiro Kiyotaki, and Albert Queralto. Financial Crises, Bank Risk Exposure, and Government Financial Policy. Unpublished working papers, Princeton University, Zhiguo He and Arvind Krishnamurthy. A Model of Capital and Crises. Review of Economic Studies, 79(2): , T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

42 Extensions Zhiguo He and Arvind Krishnamurthy. Intermediary Asset Pricing. American Economic Review, 103(2): , Nobuhiro Kiyotaki and John Moore. Credit Cycles. Journal of Political Economy, 105(2): , T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

43 Empirical Evidence Outline Empirical Evidence Additional Results Risk-Averse Intermediaries T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

44 Empirical Evidence Broker-Dealer Balance Sheets: Levels Security Broker-Dealer: Assets, Liabilies, Equity, Leverage Dollars (trillions) Assets to Equity Source: Flow of Funds Total Liabilities (left axis) Total Assets (left axis) Leverage (right axis) Back T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

45 Empirical Evidence Broker-Dealer Balance Sheets: Annual Growth Security Broker-Dealer: Assets, Liabilies, Equity, Leverage Percent Percent Total Assets Equity Total Liabilities Leverage Source: Flow of Funds Back T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

46 Empirical Evidence Broker-Dealer Balance Sheets: Adjustments Security Broker Dealer Change of Assets as a Function of Change in Equity and in Liabilities (Annual) Liabilities Changes Equity Changes Equity Growth Issuance Growth Source: Federal Reserve Flow of Funds. Equity is at book value Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07 Jan-09 Back T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

47 Empirical Evidence Balance Sheet Adjustments 300 Investment Banks 800 Commercial Banks (FDIC) Change in Equity & Changes in Debt (Billions) Equity Debt Change in Equity & Changes in Debt (Billions) Equity Debt Change in Assets (Billions) Change in Assets (Billions) Back T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

48 Empirical Evidence Broker-Dealers and Banks Back T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

49 Empirical Evidence Broker-Dealer VaR T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

50 Empirical Evidence Broker-Dealer VaR Back T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

51 Additional Results Outline Empirical Evidence Additional Results Risk-Averse Intermediaries T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

52 Additional Results Simulation Parameters Parameter Value ā σ a ρ 0.06 ρ h σξ 2 / φ φ 1 20 λ k 0.03 ρ ξ,a 0 σ ξ α 2.5 Ref.: Brunnermeier and Sannikov (2012) Monthly simulation frequency paths; 70 years Back T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

53 Additional Results Equilibrium Lemma 1 µ Rk,t = K 0 (ω t, θ t ) + K a (ω t, θ t ) σ ka,t + σ ξ 1 ρ 2 ξ,a σ kξ,t µ Rb,t = B 0 (ω t, θ t ) + B a (ω t, θ t ) σ ka,t + B ξ (ω t, θ t ) σ kξ,t µ ωt = O 0 (ω t, θ t ) + O a (ω t, θ t ) σ ka,t + O ξ (ω t, θ t ) σ kξ,t µ θt = S 0 (ω t, θ t ) + S a (ω t, θ t ) σ ka,t O ξ (ω t, θ t ) σ kξ,t r ft = R 0 (ω t, θ t ) + R a (ω t, θ t ) σ ka,t σ ba,t = 2θ tω t p kt + β (1 ω t ) σ a 2θ tω t p kt + β (1 θ t ω t ) σ ka,t βω t (θ t 1) βω t (θ t 1) σ bξ,t = 2θ tω t p kt + β (1 θ t ω t ) σ kξ,t βω t (θ t 1) σ θa,t = 2θ tω t p kt + β (1 ω t ) (σ ka,t σ a ) βω t σ θξ,t = 2θ tω t p kt + β (1 ω t ) σ kξ,t βω t θt 2 σ kξ,t = α 2 σ2 ka,t ( ) σ ka,t = θ 2 t α ω t σ2 a 1 +. ω t (2θ t ω t p kt + β (1 ω t )) T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

54 Additional Results Risk-free Rate Household Euler equation ( ) r ft = ρ h σ2 ξ 2 Goods market clearing implies dc t = d (K t A t i t k t A t ) + 1 [ ] dt E dct [ dc 1dt c E t 2 t ct 2 + dc t, dξ t 2 c t = A t dk t + (K t i t k t ) da t A t k t di t A t i t dk t k t di t, da t ] T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

55 Additional Results Stress Tests Inherent limitations to VaR include [... ] VaR does not estimate potential losses over longer time horizons where moves may be extreme. T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

56 Additional Results Stress Tests Could consider a forward-looking capital constraint [ T ( ) ] θt 1 ϑ E t σka,s 2 + σ2 kξ,s ds. Looks like a robust-control constraint Rewrite intermediary optimization as s.t. V t (ϑ) = max {i,β,k,α s} E t θ 1 s α s θ 1 t Choose optimal capital plan t [ τd t ] e ρ(s t) w t (i, β, k) ds σka,s 2 + σ2 kξ,s [ T θ 2 ] s = ϑ E t t αs 2 ds. Back T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

57 Risk-Averse Intermediaries Outline Empirical Evidence Additional Results Risk-Averse Intermediaries T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

58 Risk-Averse Intermediaries Intermediaries Banks Create new capital; financed by debt issuance to the households Funds Hold existing capital; financed by profit sharing contracts with households Two types of intermediaries: non-bank ( fund ) and bank Unit mass of specialists manage funds; unit mass of bankers manage banks Future work: interactions between different intermediary types T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

59 Risk-Averse Intermediaries Fund Sector Modeled as in He and Krishnamurthy (2012, 2013) Fund is formed each period t as a random match between a specialist and a household Specialist contributes all of his wealth w ft to the fund Household contributes up to mw ft to the fund m: tightness of the specialists capital constraint Specialists control the allocation of fund capital to holding capital projects and risk-free debt Notice: No new capital project creation No risky debt Back T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

60 Risk-Averse Intermediaries Specialists Optimization I Specialists maximize expected consumption [ + ] max E e ρt log c ft dt, {c ft,θ ft } 0 subject to the dynamic budget constraint dw ft w ft = θ ft (dr kt r ft dt) + r ft dt c ft w ft dt T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

61 Risk-Averse Intermediaries Specialists Optimization II Lemma 2 The specialists consume a constant fraction of their wealth c ft = ρw ft, and allocate the fund s capital as a mean-variance investor θ ft = µ Rk,t r ft σka,t 2 +. σ2 kξ,t Back T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

62 Risk-Averse Intermediaries Banking Sector Banks create new capital dk t = (Φ(i t ) λ k ) k t dt Investment carries quadratic adjustment costs (Brunnermeier and Sannikov (2012)) ( ) Φ (i t ) = φ φ1 i t 1 Banks finance investment projects through inside equity and outside risky debt giving the budget constraint p kt A t k t = p bt A t b t + w t Back T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

63 Risk-Averse Intermediaries Bankers Optimization I The representative banker solves dw t w t = θ t max θ t,i t,c bt [ + ] E e ρt log c bt dt 0 subject to the dynamic budget constraint ( dr kt r ft dt + and the risk-based capital constraint ( Φ (i t ) i t p kt ) ) dt (θ 1) (dr bt r ft dt) + r ft dt c bt w t dt, θ t 1. α σka,t 2 + σ2 kξ,t T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

64 Risk-Averse Intermediaries Bankers Optimization II Lemma 3 The representative banker optimally consumes at rate c bt = ρw t and invests in new projects at rate i t = 1 ( φ 2 0 φ 2 ) 1 φ 1 4 p2 kt 1. While the capital constraint in not binding, the banking system leverage is θ t = σ2 ba,t σ ka,tσ ba,t + σ 2 bξ,t σ kξ,tσ bξ,t (µ Rb,t r ft ) ((σ ba,t σ ka,t ) 2 + (σ bξ,t σ kξ,t ) 2) ( ) µ Rk,t + Φ (i t ) it p kt r ft + ((σ ba,t σ ka,t ) 2 + (σ bξ,t σ kξ,t ) 2). T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

65 Risk-Averse Intermediaries Households Back Household preferences are: E [ + 0 ] e (ξt+ρht) log c t dt Liquidity preference shocks (as in Allen and Gale (1994) and Diamond and Dybvig (1983)) are exp ( ξ t ) dξ t = σ ξ dz ξt Households allocate wealth between risky bank debt and contributions to funds T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

66 Risk-Averse Intermediaries Households Optimization I The representative household solves max π kt,π bt c t E [ + subject to the dynamic budget constraint dw ht w ht 0 ] e ξt ρht log c t dt, = π kt θ ft (dr kt r ft dt) + π bt (dr bt r ft dt) + r ft dt c t w ht dt, the skin-in-the-game constraint and no shorting constraints π kt w ht mw ft, π kt 0 b ht 0. T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

67 Risk-Averse Intermediaries Households Optimization II Lemma 4 The households optimal consumption choice satisfies ( ) c t = ρ h σ2 ξ 2 While the households are unconstrained in their wealth allocation, the households optimal portfolio choice is given by [ πkt π bt ] ([ θft σ = ka,t θ ft σ kξ,t σ ba,t σ bξ,t [ ] 1 [ ] θft σ ka,t σ ba,t 0. θ ft σ kξ,t σ bξ,t σ ξ w ht. ] [ ]) 1 [ ] θft σ ka,t σ ba,t θft (µ Rk,t r ft ) θ ft σ kξ,t σ bξ,t µ Rb,t r ft Back T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,

68 Risk-Averse Intermediaries Equilibrium An equilibrium in the economy is a set of price processes {p kt, p bt, r ft} t 0, a set of household decisions {π kt, b ht, c t} t 0, a set of specialist decisions {k ft, c ft} t 0, and a set of intermediary decisions {k t, i t, b t, c bt} t 0 such that the following apply: 1 Taking the price processes, the specialist decisions and the intermediary decisions as given, the household s choices solve the household s optimization problem, subject to the household budget constraint, the no shorting constraints and the skin-in-the-game constraint for the funds. 2 Taking the price processes, the specialist decisions and the household decisions as given, the intermediary s choices solve the intermediary s optimization problem, subject to the intermediary budget constraint, and the regulatory constraint. 3 Taking the price processes, the household decisions and the intermediary decisions as given, the specialist s choices solve the specialist s optimization problem, subject to the specialist budget constraint. 4 The capital market clears at all dates 5 The risky bond market clears 6 The risk-free debt market clears 7 The goods market clears k t + k ft = K t. b t = b ht. w t + w ft + w ht = p kta tk t. c t + c bt + c ft + A tk ti t = K ta t. Back T. Adrian, N. Boyarchenko Intermediary Leverage Cycles January,