A Macroeconomic Framework for Quantifying Systemic Risk Zhiguo He, University of Chicago and NBER Arvind Krishnamurthy, Northwestern University and NBER November 2012 He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November 2012 1 / 36
Financial Crisis in the Model 7 Sharpe ratio 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 scaled intermediary reputation e Note: Capital constraint binds for e < 0.44 He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November 2012 2 / 36
Objective Develop a macroeconomic model to evaluate systemic risk. Systemic risk: risk (probability) of a state where financial intermediation is disrupted Why is this valuable? Can ask, what is the probability of systemic states based on an initial condition. Can ask how that probability will increase given an assumed stress scenario: Translate stress tests into systemic risk probabilities Can integrate macro models with more statistical approaches to predicting crises. He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November 2012 3 / 36
Outline 1 Nonlinearity in model and data The crisis state affects non-crisis states through anticipation effects Match conditional moments of the data, conditioning on negative (i.e., recession), but not crisis states 2 Extrapolate the model to state where constraint is binding to see how well a model with shocks only to financial intermediary balance sheets matches 2007-2009 events. 3 What is the probability of systemic risk (crisis states)? 4 Stress testing in the model, and systemic risk probability. He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November 2012 4 / 36
Innovation Relative to Much of Literature We study a model with occasionally binding financial constraint Typical models (e.g., Kiyotaki-Moore (1997),...) linearize around steady state where constraint binds. 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 Main drawback: need to reduce state variables Have to leave out some common DSGE elements Similar methodology to Mendoza (2010) and Brunnermeier-Sannikov (2012) Model elements adopted from He-Krishnamurthy (2011), with real investment and housing He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November 2012 5 / 36
Agents and Technology Two classes of agents: households and bankers Households:»Z E e `c ρt y 1 φ φ t ct h dt, 0 Two types of capital: productive capital K t and housing capital H. Fixed supply of housing H 1 Price of capital qt and price of housing P t determined in equilibrium He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November 2012 6 / 36
Agents and Technology Two classes of agents: households and bankers Households:»Z E e `c ρt y 1 φ φ t ct h dt, 0 Two types of capital: productive capital K t and housing capital H. Fixed supply of housing H 1 Price of capital qt and price of housing P t determined in equilibrium Production Y = AK t, with A being constant Fundamental shocks: stochastic capital quality shock dz t dk t K t = i tdt δdt + σdz t He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November 2012 6 / 36
Agents and Technology Two classes of agents: households and bankers Households:»Z E e `c ρt y 1 φ φ t ct h dt, 0 Two types of capital: productive capital K t and housing capital H. Fixed supply of housing H 1 Price of capital qt and price of housing P t determined in equilibrium Production Y = AK t, with A being constant Fundamental shocks: stochastic capital quality shock dz t dk t K t = i tdt δdt + σdz t Investment/Capital i t, quadratic adjustment cost Φ(i t, K t) = i tk t + κ 2 (it δ)2 K t max i t q ti tk t Φ(i t, K t) i t = δ + qt 1 κ He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November 2012 6 / 36
Aggregate Balance Sheet Loans to Capital Producers i t Intermediary Sector Household Sector Capital q tk t Equity E t Financial Wealth W t = q tk t + p th Housing P th Debt W t E t He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November 2012 7 / 36
Aggregate Balance Sheet Loans to Capital Producers i t Intermediary Sector Household Sector Capital q tk t Housing P th Equity E t Debt W t E t Financial Wealth W t = q tk t + p th (1 λ)w t λw t = "Liquid balances" He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November 2012 8 / 36
Aggregate Balance Sheet Loans to Capital Aggregate bank reputation E t Producers i t Intermediary Sector Household Sector Capital q tk t Housing P th Equity E t Constraint: E t E t No constraint Debt W t E t Financial Wealth W t = q tk t + p th (1 λ)w t λw t He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November 2012 9 / 36
Single Bank/Banker Capital q tk t Housing P th t equity t debt t Portfolio share in capital: α k t = q tk t equity t Portfolio share in housing : α h t = P th t equity t Borrowing (no constraint): debt t = q tk t + P th t equity t = (α k t + α h t 1)equity t Return on bank equity: d R t = α k t dr k t + α h t dr h t (α k t + α h t 1)r tdt Banker (log preference) solves: max α k t,α h t E t[d R t r tdt] m 2 Vart[d R t] He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 10 / 36
Single Bank/Banker Capital q tk t Housing P th t equity t debt t Properties (k, h) scales with equity (k, h) increasing in E t[dr r] (k, h) decreasing in Var t[dr] Portfolio share in capital: α k t = q tk t equity t Portfolio share in housing : α h t = P th t equity t Borrowing (no constraint): debt t = q tk t + P th t equity t = (α k t + α h t 1)equity t Return on bank equity: d R t = α k t dr k t + α h t dr h t (α k t + α h t 1)r tdt Banker (log preference) solves: max α k t,α h t E t[d R t r tdt] m 2 Vart[d R t] He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 11 / 36
General Equilibrium (1) Intermediary Sector Household Sector Capital q tk t Equity E t 2 Financial Wealth Housing p th Debt W t E t Constraint: E t E t W t = q tk t + p th Portfolio share in capital: α k t = q t K t E t = q t K t min[e t,(1 λ)w t ] Portfolio share in housing: α h t = P t H E t = q t K t min[e t,(1 λ)w t ] Given state (K t, E t), the equilibrium portfolio shares are pinned down by GE But portfolio shares must also be optimally chosen by banks, pinning down prices max α k t,αh t E t[d R t r tdt] m 2 Vart[d R t] Asset prices affect real side through investment He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 12 / 36
General Equilibrium (2) Intermediary Sector Household Sector Capital q tk t Equity E t 2 Financial Wealth Housing p th Debt W t E t Constraint: E t E t W t = q tk t + p th Portfolio share in capital: α k t = q t K t E t = q t K t min[e t,(1 λ)w t ] Portfolio share in housing: α k t = q tk t E t = q t K t min[e t,(1 λ)w t ] Prices (returns) have to adjust for optimality: Et[dR h t r tdt], E t[dr k t r tdt] equations for E t[dp t], E t[dq t] Rewrite to get diff eqns for P(K, E) and q(k, E) Scale invariance: Define e E/K; then P = Kp(e) and q(e), PDEs become ODEs He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 13 / 36
Capital Constraint Single bank has reputation" (skill, etc.) ɛ t linked to intermediary performance (constant m) dɛ t ɛ t = md R t. Poor returns reduce reputation: Flow-performance relationship (Chevalier-Ellison, 1997) Or, ɛt can be interpreted as net worth" of bankers fluctuating with performance (He-Krishnamurthy 2011, 2012) Households invest a maximum of ɛ t dollars of equity capital with this banker He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 14 / 36
Capital Constraint Single bank has reputation" (skill, etc.) ɛ t linked to intermediary performance (constant m) dɛ t ɛ t = md R t. Poor returns reduce reputation: Flow-performance relationship (Chevalier-Ellison, 1997) Or, ɛt can be interpreted as net worth" of bankers fluctuating with performance (He-Krishnamurthy 2011, 2012) Households invest a maximum of ɛ t dollars of equity capital with this banker E t: aggregate reputation. Aggregate dynamics of E t de t E t = md R t ηdt + dψ t Death rate η, and entry dψ t > 0 of new bankers in extreme bad states He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 14 / 36
Equity Capital Constraint Benchmark capital structure: λw t of Debt, (1 λ)w t of Equity. if there is no capital constraint (Et is infinite)... He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 15 / 36
Equity Capital Constraint Benchmark capital structure: λw t of Debt, (1 λ)w t of Equity. if there is no capital constraint (Et is infinite)... Intermediary equity capital: E t = min [E t,(1 λ)w t] Suppose a 10% shock to real estate and price of capital: W t 10% (Household wealth = aggregate wealth) Reputation: de t E t = md R t +... Two forces make E t more than 10%: 1 Equity is levered claim on assets: Return on equity = d R t < 10% 2 m > 1 in our calibration. He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 15 / 36
Micro foundation of Capital Constraint We develop theory in He-Krishnamurthy (2012, Restud), and applied to MBS market in He-Krishnamurthy (2012, AER forthcoming) Two-agents endowment economy, Households with wealth Wt h cannot hold MBS assets but can delegate their money to Bankers with wealth W t With agency friction, households are only willing to contribute at most mw t as outside equity capital, so risk-sharing rule cannot fall below 1 m "Skin in the game" idea When banker s net worth W t is low, capital constraint is binding Binding capital constraint is a binding Incentive Compatibility constraint in delegation/agency contracting problem IC binds after a series of bad shocks where banker s net worth Wt is low Banker s net worth W t evolves with fund performance, just like reputation ɛ t The mechanism of reputation-based-capital-constraint is similar He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 16 / 36
Modeling details (1): Bankers Banker die at a Poisson rate of η, which is effective discount rate Banker makes portfolio decision to maximize his/her present value of expected reputation: E [ e ηt ln ɛ 0 t dt ] Bankers do not consume goods, representative household framework Convenient to get interest rate determined solely by households With log preferences, banker chooses portfolios (for intermediary) max α k t,αh t E t [d R t r t dt] m 2 Var t[d R t ] m parameterizes banker s risk aversion He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 17 / 36
Modeling details (2): Deposit demand Representative household enters time t with financial wealth W t The household splits wealth: (1 λ) W t to equity households, λw t to bond households Equity households have portfolio choice: invest their portion of wealth as equity of intermediaries, subject to capital frictions, or in debt. Bond households only invest in riskless bonds This gives minimum holding of bank debt of λwt Once returns are realized, both members pool their wealth again (as in Lucas 1990) He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 18 / 36
Equilibrium and State Variables The definition of equilibrium is standard Markov equilibrium with state variables (E t, K t) Scale invariance. Unidimensional state variable e t = Et K t, with endogenous evolution de t = µ edt + σ edz t φ Housing price P = Kp (e), as housing rental Ct is scaled with the economy 1 φ We solve for q (e) and p (e) as solutions to a system of ODEs He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 19 / 36
Boundary Conditions When e =, E t > (1 λ) W t always, frictionless economy 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) Entry increases aggregate E but requires physical capital K at conversion rate of β e is a reflecting boundary Boundary conditions at the entry point e q (e) = 0, p (e) = p (e)β, and Sharpe_Ratio (e) = γ 1 + eβ He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 20 / 36
Calibration: Baseline Parameters Parameter Choice Targets (Unconditional) Panel A: Intermediation m Performance sensitivity 2 Average Sharpe ratio λ Debt ratio 0.67 Average intermediary leverage η Banker exit rate 17% Good model dynamics γ Entry trigger 6.5 Highest Sharpe ratio β Entry cost 2.34 Average land price volatility Panel B: Technology σ Capital quality shock 4% Consumption volatility δ Depreciation rate 10% Literature κ Adjustment cost 3 Literature A Productivity 0.148 Investment-to-capital ratio Panel C: Others ρ Time discount rate 3% Literature φ Housing share 0.5 Housing-to-wealth ratio He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 21 / 36
Results(1): State variable is e t = Et K t 7 Sharpe ratio 0.1 interest rate 6 0.05 5 0 4 3 0.05 2 0.1 1 0.15 1.03 0 0 1 2 3 4 5 6 7 8 9 10 scaled intermediary reputation e q(e), capital price 0 2 4 6 8 10 scaled intermediary reputation e investment I/K 0.105 1.02 1.01 1 0.1 0.99 0.98 0.97 0.095 0.96 0.95 0 1 2 3 4 5 6 7 8 9 10 scaled intermediary reputation e 0.09 0 2 4 6 8 10 scaled intermediary reputation e Capital constraint binds for e < 0.44 He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 22 / 36
Results(2) 0.9 p(e), scaled housing price 1.03 q(e), capital price 0.8 1.02 0.7 1.01 0.6 1 0.99 0.5 0.98 0.4 0.97 0.3 0.96 0.2 0 1 2 3 4 5 6 7 8 9 10 1.2 1 0.8 0.6 0.4 return volatility of housing 0.95 0 1 2 3 4 5 6 7 8 9 10 0.03 0.025 0.02 0.015 0.01 steady state distribution 0.2 0.005 0 0 1 2 3 4 5 6 7 8 9 10 scaled intermediary reputation e 0 0 1 2 3 4 5 6 7 8 9 10 scaled intermediary reputation e Capital constraint binds for e < 0.44 Without the possibility of the capital constraint, all of these lines would be flat. Model dynamics would be i.i.d., with vol=4% He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 23 / 36
Impulse Response: -2% Shock (= σdz t ) VARdata 0.022 Investment 0.7 Sharpe ratio 0.04 Land price 0.023 crisis normal 0.6 crisis normal 0.06 crisis normal 0.024 0.5 0.08 0.025 0.4 0.1 0.026 0.3 0.12 0.027 0.2 0.028 0.1 0.14 0.029 0 2 4 6 8 quarter 0 0 2 4 6 8 quarter 0.16 0 2 4 6 8 quarter He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 24 / 36
Nonlinearities in Model and Data Model: Data: Distress states = worst 33% of realizations of e (e < 2.14) Compute variances, covariances of intermediary equity growth with other key variables Distress states = worst 33% of realizations of credit spread We use Gilchrist-Zakrajsek (2011) as credit spread, which we convert to a Sharpe ratio Compute variances, covariances of intermediary equity growth with other key variables He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 25 / 36
Distress Classification Distress Periods NBER Recessions 1974Q3-1975Q4 11/73-3/75 1982Q3-1982Q4 7/81-11/82 1985Q4-1987Q3 1988Q4-1990Q1 7/90-3/91 1992Q4-1993Q2 2001Q2-2003Q1 3/01-11/01 2007Q3-2009Q3 12/07-6/09 He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 26 / 36
Covariances in Data EB NBER Recession NBER+,-2Qs EB, Drop Crisis Panel A: Distress Periods vol(eq) 31.48 32.40 31.78 22.19 vol(i) 8.05 8.79 7.44 4.56 vol(c) 1.71 1.54 1.59 0.95 vol(pl) 21.24 23.34 21.07 7.91 vol(eb) 60.14 93.59 74.57 28.69 cov(eq, I) 1.31 1.08 0.84 0.37 cov(eq, C) 0.25 0.16 0.13 0.04 cov(eq, PL) 4.06 5.61 4.39-0.63 cov(eq, EB) -6.81-10.89-7.57-2.12 Panel B: Non-distress Periods vol(eq) 17.54 19.42 17.11 17.26 vol(i) 6.61 5.97 4.91 6.60 vol(c) 1.28 0.98 0.91 1.28 vol(pl) 9.79 10.00 8.46 9.34 vol(eb) 12.72 30.93 30.42 12.78 cov(eq, I) 0.07 0.09-0.06 0.03 cov(eq, C) 0.03 0.01 0.01 0.03 cov(eq, PL) 0.12 0.07-0.31-0.01 cov(eq, EB) -0.14-0.81-0.78-0.19 He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 27 / 36
Matching State-Dependent Covariances Distress Non Distress Data Baseline Data Baseline vol (Eq) 31.48% 31.2 17.54 6.4 vol (I) 8.05% 5.4 6.61 4.8 vol (C) 1.71% 1.8 1.28 2.4 vol (LP) 21.24% 22.1 9.79 9.8 vol (EB) 60.14% 71.1 12.72 8.7 cov (Eq, I) 1.31% 0.90 0.07 0.3 cov (Eq, C) 0.25% 0.0 0.03 0.1 cov (Eq, LP) 4.06% 5.6 0.12 0.6 cov (Eq, EB) -6.81% -13.0-0.14-0.2 Note: without the capital constraint, all volatilities would be 4%, and have no state dependence. He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 28 / 36
Matching State-Dependent Covariances: No Housing Distress Non Distress Data Baseline φ = 0 Data Baseline φ = 0 vol (Eq) 31.48% 31.2 23.3 17.54 6.4 4.0 vol (I) 8.05% 5.4 4.7 6.61 4.8 4.3 vol (C) 1.71% 1.8 2.6 1.28 2.4 3.8 vol (LP) 21.24% 22.1 9.79 9.8 vol (EB) 60.14% 71.1 31.3 12.72 8.7 0.2 cov (Eq, I) 1.31% 0.9 0.7 0.07 0.3 0.2 cov (Eq, C) 0.25% 0.0 0.2 0.03 0.1 0.2 cov (Eq, LP) 4.06% 5.6 0.12 0.6 cov (Eq, EB) -6.81% -13.0-2.7-0.14-0.2 0.0 He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 29 / 36
Matching Recent Crisis!"! Based on realized equity return we uncover fundamental shocks to K 07QIII 07QIV 08QI 08QII 08QIII 08QIV 09QI 09QII 09QIII 09QIV -3.1% -5.5-3.0-1.4-0.8-2.2-2.3-2.2-1.0-1.0 Total -19%. Capital constraint binds after 08QI systemic crisis In the model (data), land price falls by 56% (55%) He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 30 / 36
Probability of Systemic Event Starting from 2007Q2, probability of hitting crisis 1.12% in next 2 years 9.12% in next 5 years 20.73% in next 10 years He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 31 / 36
Probability of Systemic Event Starting from 2007Q2, probability of hitting crisis Small... 1.12% in next 2 years 9.12% in next 5 years 20.73% in next 10 years VIX was low early 2007 Financial sector aggregate leverage fixed at 3 in model Currently, -13% shocks move you to crisis boundary. If leverage was higher, required shocks would be smaller... "Hidden" leverage: leverage was 4.5 but agents take as given price functions and returns at leverage=3 Due to nonlinearilty, prob of hitting crisis rises to 60%! He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 31 / 36
Stress testing Input: Output: Description of a relevant scenario Loss in terms of equity of the financial sector (ROE). How can a model help: General equilibrium response to equity loss (Brunnermeier, Gorton and Krishnamurthy (2011)) Mapping scenario into probability of systemic states He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 32 / 36
Stress testing example Key step: Need to map from stress scenario into underlying shock, dz t. Naive partial eqbm approach. Suppose stress scenario -30% Return on equity With leverage of 3, σ(z t+0.25 Z t) = 30/3 = 10%. Feed in 10% shock into the model over one quarter. Result: Beginning at e = 2.14 in 2007Q2, economy is immediately moved into crisis region, e < 0.44 our model helps in figuring out the right shock dz t In US stress tests, scenario was over 6 quarters. Feed in shocks quarter-by-quarter, over 6 quarters: Return on Equity 6 QTR Shocks Prob(Crisis within next 2 years) -2% -1.52% 1.53 % -5-3.11% 2.80-10 -5.67% 7.37-20 -10.41% 36.78-30 -13.06% 100.00 He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 33 / 36
Probability of Systemic Event Probability of capital constraint being binding 1 0.8 in next 2 years in next 5 years in next 10 years 0.6 0.4 0.2 0 0 1 2 3 4 starting value e init He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 34 / 36
Conclusion We develop a fully stochastic model of a systemic crisis, with an equity capital constraint on the intermediary sector The model quantitatively matches the differential comovements in distress and non-distress periods Is able to replicate 2007/2008 period with only intermediary capital shocks Offers a way of mapping macro-stress tests into probability of systemic states. He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 35 / 36
Nonlinearity: VAR in data Panel IR A: Distress Periods 0.2 Equity to Equity EBP (credit risk premium) to Equity 0 0.05 Investment to Equity 0.15 1 0.04 0.1 2 3 0.03 0.02 0.05 4 0.01 0 0 2 4 6 8 5 0 2 4 6 8 0 0 2 4 6 8 Panel B: Non Distress Periods 0.14 0.12 0.1 0.08 0.06 0.04 0.02 Equity to Equity 0 0 2 4 6 8 0 1 2 3 4 EBP (credit risk premium) to Equity 5 0 2 4 6 8 Investment to Equity 0.05 0.04 0.03 0.02 0.01 0 0 2 4 6 8 He and Krishnamurthy (Chicago, Northwestern) Systemic Risk November2012 36 / 36