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A Macroeconomic Framework for Quantifying Systemic Risk Zhiguo He University of Chicago & NBER Arvind Krishnamurthy Northwestern University & NBER June 2012 Systemic Risk I Systemic risk: risk (probability) of a state where I I I …nancial intermediation is disrupted small fundamental shocks to …nancial intermediaries can have quantitatively large e¤ects on macro economy Goal: Write down a non-linear macro model to assess systemic risk I I much of the time the link between …nancial intermediation and macro economy is small but in (crisis) states the e¤ects are greatly ampli…ed Systemic Risk I Systemic risk: risk (probability) of a state where I I I …nancial intermediation is disrupted small fundamental shocks to …nancial intermediaries can have quantitatively large e¤ects on macro economy Goal: Write down a non-linear macro model to assess systemic risk I I much of the time the link between …nancial intermediation and macro economy is small but in (crisis) states the e¤ects are greatly ampli…ed I How well does the model match asymmetry (i.e. occasional e¤ects of …nancial intermediation) in the data? I How well can an intermediary shock channel explain patterns in 2007-2009? I How likely is the economy, say unconditionally, to enter a systemic risk episode? Innovation Relative to Much of Literature I We study a model with occasionally binding …nancial constraint I Typical models (e.g., Kiyotaki-Moore (1997),...) linearize around steady state where constraint binds. I I Cannot talk about 1) likelihood that intermediation is disrupted (its always disrupted...) and 2) how severely it is disrupted Our model solution has stochastic steady state, with fully solved equilibrium prices and policies I I Main drawback: need to reduce state variables Have to leave out some common DSGE elements I Similar methodology to Mendoza (2010) and Brunnermeier-Sannikov (2011) I Model elements adopted from He-Krishnamurthy (2012), with real investment and housing Preview of model result Sharpe ratio interest rate 6 0.1 5 0.05 0 4 -0.05 3 -0.1 2 -0.15 1 0 0 -0.2 5 10 scaled intermediary reputation e 15 20 -0.25 0 5 investment I/K 10 scaled intermediary reputation e 15 20 steady state distribution 0.105 0.035 0.104 0.03 0.103 0.025 0.102 0.101 0.02 0.1 0.015 0.099 0.01 0.098 0.005 0.097 0.096 0 5 10 scaled intermediary reputation e 15 20 0 0 5 10 15 20 25 scaled intermediary reputation e 30 35 40 Preview of model result Sharpe ratio interest rate 6 0.1 5 0.05 0 4 -0.05 3 -0.1 2 -0.15 1 0 0 -0.2 5 10 scaled intermediary reputation e 15 20 -0.25 0 5 investment I/K 10 scaled intermediary reputation e 15 20 steady state distribution 0.105 0.035 0.104 0.03 0.103 0.025 0.102 0.101 0.02 0.1 0.015 0.099 0.01 0.098 0.005 0.097 0.096 0 I I 5 10 scaled intermediary reputation e 15 20 0 0 5 10 15 20 25 scaled intermediary reputation e 30 35 Crisis: ecrisis = 0.65, binding capital constraint Distress: edistress = 4 so that Pr (e edistress ) = 33% as in data 40 Strategy I Crises are rare. How do we quantify model? I Even if economy is currently not in a crisis state, the anticipation of a crisis a¤ects decisions. 1. We match data on “distress" (33% of data) and “non-distress" periods (67% of data). 2. We extrapolate to a crisis and ask how well the model can match patterns from 2007-2009. 3. We compute conditional probabilities of triggering a crisis (measuring “systemic risk" probabilities). Evidence of Non-Linearity I Excess bond premium (EBP): the risk premium part of credit spread (removing default part), Gilchrist and Zakrajsek (2010). Correlates with measures of intermediary health. I Use EBP to classify distress periods (33%) and non-distress periods (the rest) Distress Periods 1973Q1 - 1975Q3 1982Q2 - 1982Q4 1985Q4 - 1987Q3 1988Q4 - 1990Q1 1992Q4 - 1993Q2 2001Q2 - 2003Q1 2007Q3 - 2009Q3 NBER Recessions 11/73 - 3/75 7/81 - 11/82 7/90 - 3/91 3/01 - 11/01 12/07 - 6/09 State-Dependent Covariances (1) I Equity = Total market value of equity of …nance, insurance and real estate sectors. (works as well if only include banks + broker/dealers) I All variables are growth, except Sharpe ratio constructed from EBP Distress Non Distress Cov Corr Cov Corr Equity , Investment 1.31% 51.48 0.07 5.79 Equity , Consumption 0.25% 45.85 0.03 14.74 Equity , Sharpe -6.81% -35.96 -0.14 -0.06 Equity , Landprice 4.06% 60.65 0.12 0.07 State-Dependent Covariances (2) All variables are growth, except Sharpe ratio constructed from EBP Distress Non Distress NBER+2 Excl-Crisis NBER+2 Excl-Crisis Equity , Investment 0.84% 0.37 -0.06 0.03 Equity , Consumption 0.13% 0.04 0.01 0.03 Equity , Sharpe -7.57% -2.12 -0.78 -0.19 Equity , Landprice 4.39% -0.63 -0.31 -0.01 Note: Similar numbers if only use NBER dates, but distress sample is only 20% of observations. VAR Evidence of Non-Linearity (3) VAR order: [intermediary equity, aggregate stock market, EBP, investment]. Coe¢ cients depend on distress/non-distress state. Quarterly growth rates. PANEL A: DISTRESS PERIODS Market to Equity Equity to Equity 25 EB (credit risk premium) to Equity 0 20 Investment to Equity 5 -0.2 20 15 4 -0.4 15 3 -0.6 10 2 -0.8 10 5 5 1 -1 0 -1.2 0 0 2 4 6 8 0 0 2 4 6 8 -1.4 0 2 4 6 8 -1 0 2 4 6 8 PANEL B: NON DISTRESS PERIODS Equity to Equity Market to Equity 20 Investment to Equity EB to Equity 12 0.2 2 10 0.1 1.5 15 0 8 1 -0.1 10 6 0.5 -0.2 4 2 0 0 2 4 6 8 0 0 -0.3 5 -0.5 -0.4 0 2 4 6 8 -0.5 0 2 4 6 8 -1 0 2 4 6 8 Road Map of the Rest of Talk I Model, mechanism, and solution I Calibration I I Baseline parameters Prices and polices, comparative statics I Matching data on distress and non-distress I Systemic crisis I I I Extrapolate to crisis state Uncover fundamental shocks in the recent crisis How likely are crises? Agents and Technology I Two classes of agents: households and bankers I I Households own the entire economy, but subject to frictions related to bankers who control intermediaries (next slide) Two types of capital: productive capital Kt and housing capital H. Fixed supply of housing H 1 I Price of capital qt and price of housing Pt determined in equilibrium Agents and Technology I Two classes of agents: households and bankers I I Households own the entire economy, but subject to frictions related to bankers who control intermediaries (next slide) Two types of capital: productive capital Kt and housing capital H. Fixed supply of housing H 1 I Price of capital qt and price of housing Pt determined in equilibrium I Production Y = AKt , with A being constant I Fundamental shocks: stochastic capital quality shock dZt dKt = it dt Kt I δdt + σdZt Investment/Capital it , quadratic adjustment cost Φ(it , Kt ) = it Kt + κ (it 2 δ)2 Kt Aggregate Balance Sheet ' Loans to Capital Producers it $ & % 6 Intermediary Sector Capital qt Kt Housing Pt H Household Sector HH Y HH Equity Et Debt Wt Et Financial Wealth HH Wt = qt Kt + Pt H Aggregate Balance Sheet ' Loans to Capital Producers it $ Aggregate bank reputation Et & % 6 Intermediary Sector Capital qt Kt Equity Et Household Sector HH Y Constraint:HEH t Financial Wealth Et HH Wt = qt Kt + Pt H No constraint Housing Pt H Debt Wt Et Single Bank/Banker Capital qt kt Housing Pt ht Equity et Debt dt Portfolio share in capital: αkt = qet kt t Portfolio share in housing : αht = Pet tht Borrowing (no constraint): dt = qt kt + Pt ht et = (αkt + αht Return on bank equity: d R̃t = αkt dRtk + αht dRth (αkt + αht Banker (log preference) solves: maxαk ,αh E [d R̃t t t rt dt ] 1)rt dt m Var [d R̃ ] t t 2 1 ) et Single Bank/Banker Capital qt kt Housing Pt ht Equity et Debt dt Properties (k, h) scales with e (k, h) increasing in Et [dR r ] (k, h) decreasing in Var [dR ] Portfolio share in capital: αkt = qet kt t Portfolio share in housing : αht = Pet tht Borrowing (no constraint): dt = qt kt + Pt ht et = (αkt + αht Return on bank equity: d R̃t = αkt dRtk + αht dRth (αkt + αht Banker (log preference) solves: maxαk ,αh E [d R̃t t t rt dt ] 1)rt dt m Var [d R̃ ] t t 2 1 ) et General Equilibrium (1) Intermediary Sector Capital qt Kt Equity Et Housing pt H Debt Wt Household Sector Financial Wealth yXX X Constraint: Et XXXEt XXX Et Wt = qt Kt + pt H Portfolio share in capital: αkt = qEt Kt t Portfolio share in housing: αht = PEt tH I Given a particular state (Kt , Et ), the portfolio shares are pinned down by GE I Portfolio shares must also be optimally chosen by banks max Et [d R̃ t αkt ,αht rt dt ] m Vart [d R̃ t ] 2 General Equilibrium (2) Intermediary Sector Capital qt Kt Equity Et Housing pt H Debt Wt Household Sector Financial Wealth yXX X Constraint: E XX Et XXX t X W = q K +p H Et t t t t Portfolio share in capital: αkt = qEt Kt t Portfolio share in housing: αht = PE tt I Prices (returns) have to adjust for optimality: I I I Et [dRth rt dt ], Et [dRtk rt dt ] ) equations for Et [dPt ], Et [dqt ] Rewrite to get ODEs for P (K , E ) and q (K , E ) Scale invariance: De…ne e E /K ; then P = Kp (e ) and q (e ) Capital Producers and Investment I Capital goods producers (owned by households) undertake real investment I Producers must sell the capital stock to intermediaries at price qt I I I Possible interpretations: I I I Risk averse intermediaries bear aggregate fundamental shocks Real investment is a¤ected by …nancial condition of intermediaries to capture “credit crunch” Entrepreneurs raise capital from VC/PE at the price of qt Commercial banks makes collateralized loans Investment decision max qt it Kt it Φ(it , Kt ) ) it = δ + qt 1 κ Capital Constraint I Single bank has reputation et linked to intermediary performance (constant m) d et = mR̃t . et I I Poor past returns reduce reputation Households invest a maximum of et dollars of equity capital with this banker Capital Constraint I Single bank has reputation et linked to intermediary performance (constant m) d et = mR̃t . et I Poor past returns reduce reputation I Households invest a maximum of et dollars of equity capital with this banker I Death rate η, and entry d ψt > 0 of new bankers in extreme states (modeled later) I Et : aggregate reputation. Identical banks, aggregate dynamics of Et d Et = md R̃t Et ηdt + d ψt Capital Constraint I Single bank has reputation et linked to intermediary performance (constant m) d et = mR̃t . et I Poor past returns reduce reputation I Households invest a maximum of et dollars of equity capital with this banker I Death rate η, and entry d ψt > 0 of new bankers in extreme states (modeled later) I Et : aggregate reputation. Identical banks, aggregate dynamics of Et d Et = md R̃t Et I ηdt + d ψt Note: Et is like “net worth" in many other models. Households’ Problem (1) I Choose consumption cty and housing cth to maximize E I Z ∞ 0 e ρt (1 φ) ln cty + φ ln cth dt Equilibrium rental price Dt (housing asset dividend), FOC cth D t φ = cty 1 φ. In equilibrium (Cth = H = 1) Dt = I φ 1 Cy φ t φ: expenditure share in housing, or the relative size of housing sector I Households free to trade short-term debt. I Interest rate rt = ρ + Et dCty /Cty Vart dCty /Cty Households’ Problem (2) I Representative household enters time t with …nancial wealth Wt I The household splits wealth: (1 λWt to “bond households” I I λ) Wt to “equity households,” Equity households invest their portion of wealth as equity of intermediaries, subject to capital frictions Bond households invest in riskless bonds I Once returns are realized, both members pool their wealth again (as in Lucas 1990) I The only role of bond households (i.e. parameter λ) is to introduce intermediary’s leverage in normal time Debt/Equity Ratio ' Loans to Capital Producers it $ Aggregate bank reputation Et & % 6 Intermediary Sector Capital qt Kt Housing Pt H Equity Et Debt Wt Household Sector HH Y Constraint:HEH t Financial Wealth Wt = qt Kt + pt H Et HH ( 1 λ ) Wt No constraint Et λWt Equity Capital Constraint I I Unconstrained capital structure: λWt of Debt, (1 Intermediary equity capital Et is given by Et = min [Et , (1 I I I λ)Wt of Equity. λ ) Wt ] How can capital constraint come to bind, beginning in a state where E t > ( 1 λ ) Wt ? Suppose a 10% shock to real estate and price of capital, so that Wt # 10% (Household wealth = aggregate wealth) Reputation follows dEEt t = md R̃t + ... Two forces make Et # more than 10%: I I Equity is levered claim on assets: Return on equity = d R̃t < m > 1 in our calibration. 10% Boundary Conditions I When e = ∞, Et > (1 I I I We solve for p (∞), q (∞) analytically As e ! 0, intermediaries’portfolio volatility, i.e. Sharpe ratio, rises New bankers enter if e = e (Sharpe ratio hits γ, exogenous constant) I I I λ) Wt frictionless economy Entry increases aggregate E but requires physical capital K at conversion rate of β e is a re‡ecting boundary Boundary conditions at the entry point e q 0 (e ) = 0, p 0 (e ) = p (e ) β , and Sharpe_Ratio (e ) = γ 1 + eβ Calibration: Baseline Parameters Parameter Panel A: Intermediation m Performance sensitivity λ Debt ratio η Banker exit rate γ Entry trigger β Entry cost Panel B: Technology σ Capital quality shock δ Depreciation rate κ Adjustment cost A Productivity Panel C: Others ρ Time discount rate φ Housing share Choice Target 2.5 0.5 13% 5.5 2.35 Average Sharpe ratio (38%) Average intermediary leverage Good model dynamics Highest Sharpe ratio Land price volatility 5% 10% 2 0.14 Investment and Consumption volatilities Literature Literature Investment-to-capital ratio 2% 0.5 Literature Housing-to-wealth ratio Equilibrium Prices and Policies (1) I ecrisis = 0.65: binding capital constraint I edistress = 4 so that Pr (e edistress ) = 33% as in data p(e), s c aled hous ing pric e q(e), c apital pric e 0.8 1.01 1.008 0.7 1.006 0.6 1.004 0.5 1.002 1 0.4 0.998 0.3 0.996 0.2 0.1 0 0.994 2 4 6 8 10 12 14 scaled intermediary reputation e 16 18 20 0.992 0 2 4 return v olatility of hous ing 6 8 10 12 14 scaled intermediary reputation e 16 18 20 16 18 20 return v olatility of c apital 1.5 0.0535 0.053 0.0525 1 0.052 0.0515 0.051 0.5 0.0505 0.05 0.0495 0 0 2 4 6 8 10 12 14 scaled intermediary reputation e 16 18 20 0.049 0 2 4 6 8 10 12 14 scaled intermediary reputation e Equilibrium Prices and Policies (2) I I ecrisis = 0.65: binding capital constraint edistress = 4 so that Pr (e edistress ) = 33% as in data Sharpe ratio interest rate 6 0.1 5 0.05 0 4 -0.05 3 -0.1 2 -0.15 1 0 0 -0.2 5 10 scaled intermediary reputation e 15 20 -0.25 0 5 investment I/K 10 15 scaled intermediary reputation e 20 steady state distribution 0.105 0.035 0.104 0.03 0.103 0.025 0.102 0.101 0.02 0.1 0.015 0.099 0.01 0.098 0.005 0.097 0.096 0 5 10 scaled intermediary reputation e 15 20 0 0 5 10 15 20 25 30 scaled intermediary reputation e 35 40 Matching State-Dependent Covariances: Baseline vol (Eq ) vol (I ) vol (C ) vol (LP ) vol (EB ) cov (Eq, I ) cov (Eq, C ) cov (Eq, LP ) cov (Eq, EB ) Distress Data Baseline 31.48% 26.01 8.05% 5.73 1.71% 3.29 21.26% 22.87 60.14% 49.96 1.31% 0.80 0.25% 0.34 4.06% 4.56 -6.81% -6.69 Non Distress Data Baseline 17.54 6.77 6.61 5.39 1.28 3.94 9.79 9.38 12.72 6.32 0.07 0.36 0.03 0.26 0.12 0.63 -0.14 -0.09 Matching State-Dependent Covariances: lower σ Distress vol (Eq ) vol (I ) vol (C ) vol (LP ) vol (EB ) cov (Eq, I ) cov (Eq, C ) cov (Eq, LP ) cov (Eq, EB ) Non Distress Baseline σ = 4% Data Baseline σ = 4% 31.48% 26.01 20.57 17.54 6.77 5.09 8.05% 5.73 4.40 6.61 5.39 4.18 Data 1.71% 3.29 2.67 1.28 3.94 3.36 21.24% 21.26 12.91 9.79 9.38 6.41 60.14% 48.96 36.52 12.72 6.32 4.33 1.31% 0.81 0.47 0.07 0.37 0.22 0.25% 0.35 0.22 0.03 0.26 0.17 4.06% 4.56 2.03 0.12 0.63 0.34 -6.81% -6.69 -3.58 -0.14 -0.09 -0.04 Matching State-Dependent Covariances: No Housing Distress vol (Eq ) vol (I ) vol (C ) vol (LP ) vol (EB ) cov (Eq, I ) cov (Eq, C ) cov (Eq, LP ) cov (Eq, EB ) Non Distress Baseline φ=0 Data 31.48% 26.09 14.62 8.05% 5.73 5.14 1.71% 3.29 4.52 21.24% 21.26 60.14% 48.96 1.31% 0.25% Data Baseline φ=0 17.54 6.77 5.00 6.61 5.40 5.01 1.28 3.92 4.94 9.79 9.38 9.42 12.72 6.32 0.03 0.81 0.53 0.07 0.37 0.25 0.35 0.44 0.03 0.26 0.25 0.12 0.63 -0.33 -0.14 -0.09 4.06% 4.56 -6.81% -6.69 -0.00 Uncovering Shocks in the Recent Crisis Data Model Uncovering Shocks in the Recent Crisis Data I I Model Based on realized equity return we uncover fundamental shocks to K 07QIII 07QIV 08QI 08QII 08QIII 08QIV 09QI 09QII 09QIII 09QIV -3.77% -7.24 -6.62 -2.85 -0.48 -3.10 -2.32 -1.15 -0.04 -0.77 Total -25%. Capital constraint binds after 08QII— systemic crisis I In the model (data), land price fall by 71% (55%) Probability of Crisis I 2007Q2, Prob(crisis occurs in the next 2 years)=0.09%, Prob(5 years) = 2.62%, Prob (10 years) = 10.05% Probability of Crisis I 2007Q2, Prob(crisis occurs in the next 2 years)=0.09%, Prob(5 years) = 2.62%, Prob (10 years) = 10.05% I Conditional probability of hitting crisis (left) or distress (right) Probability of Entering D is tres s States Probability of C apital C ons traint Binding 0.35 1.1 1 0.3 0.9 0.25 0.8 in nex t 5 y ears in nex t 10 y ears 0.2 in nex t 5 y ears in nex t 10 y ears 0.7 0.6 0.15 0.5 0.4 0.1 0.3 0.05 0.2 0 0 2 4 6 8 Initial C ondition e init 10 12 0.1 0 2 4 6 8 Initial C ondition: e 10 init Note: Probabilities are low, which suggests improved capital bu¤ers would have limited e¤ects. 12 VIX and Systemic Risk e, s c aled i ntermediary reputation 20 15 10 5 0 0 50 100 150 200 250 300 tim e (quarters ) prob. of c apital c ons traint being c ons trained in nex t x y ears 350 400 450 1 nex t 1 y ear nex t 5 y ears nex t 10 y ears 0.5 0 0 50 100 150 50 100 150 200 tim e (quarters ) hous i ng pric e v olatility 250 300 350 400 250 300 350 400 0.4 0.3 0.2 0.1 0 0 200 tim e (quarters ) I Volatility in our model rises most sharply when the constraint binds I Coincident indicator and no predictive content. I What might work better? A VIX spread: Long-maturity VIX minus short-maturity VIX I Other indicators... Conclusion I We develop a fully stochastic model of systemic crisis, with two major frictions: I I I Equity capital constraint on intermediary sector Intermediaries have substantial holdings in real assets (physical capital or housing) We …nd that the model I I not only qualitatively delivers the nonlinearity observed in the data but also quantitatively matches the di¤erential comovements in distress and non-distress periods I Recent 07/08 crisis requires a cumulative negative shock around -25% I Things we are working on: more on model-based measure of systemic risk