A Macroeconomic Framework for Quantifying Systemic Risk

A Macroeconomic Framework for Quantifying Systemic Risk Zhiguo He, University of Chicago and NBER Arvind Krishnamurthy, Stanford University and NBER March 215 He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 1 / 28

Financial Crisis in the Model 7 Sharpe ratio 6 5 4 3 2 1 2 4 6 8 1 12 14 16 18 2 scaled intermediary reputation e Note: Capital constraint binds for e <.435 He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 2 / 28

Non-linearity: State-dependent Impulse Response: -1% Shock.11 Investment.45 Sharpe ratio.1 Land price.12.4.2.13.35.3.14.3.4.15.16.17 crisis normal.25.2.15.1 crisis normal.5.6.7 crisis normal.18.5.8.19 2 4 6 8 quarter 2 4 6 8 quarter.9 2 4 6 8 quarter He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 3 / 28

Global Solution: Steady State Distribution.4 steady state distribution.35.3.25.2.15.1.5 5 1 15 scaled intermediary reputation e He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 4 / 28

Model-based stress test Pick initial condition to match 27Q2 asset prices Probability of crisis over horizon: 1 year:.32% 2 year: 3.57% 5 year: 17.3 % Initial condition + rational forward looking agents = can t see around corners! He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 5 / 28

Model-based stress test Pick initial condition to match 27Q2 asset prices Probability of crisis over horizon: 1 year:.32% 2 year: 3.57% 5 year: 17.3 % Initial condition + rational forward looking agents = can t see around corners! Stress test: Add $2 trillion of shadow banking liabilities, with close to % capital. This information was not in 27Q2 asset prices Probability of crisis over horizon: 1 year: 6.73% 2 year: 23.45% 5 year: 57.95 % He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 5 / 28

Outline of Presentation 1 Nonlinear macro model of a financial crisis Recent work on financial intermediaries: He-Krishnamurthy, Brunnermeier-Sannikov, Rampini-Viswanathan, Adrian-Boyarchenko, Gertler-Kiyotaki Our approach: occasionally binding constraint; global solution method (similar to Brunnermeier-Sannikov, Adrian-Boyarchenko) He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 6 / 28

Outline of Presentation 1 Nonlinear macro model of a financial crisis Recent work on financial intermediaries: He-Krishnamurthy, Brunnermeier-Sannikov, Rampini-Viswanathan, Adrian-Boyarchenko, Gertler-Kiyotaki Our approach: occasionally binding constraint; global solution method (similar to Brunnermeier-Sannikov, Adrian-Boyarchenko) 2 Calibration and results 3 Quantify systemic risk and stress test He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 6 / 28

Model Two classes of agents: households and bankers Households:»Z E e ρt 1 1 γ C1 γ t dt, C t = `c y 1 φ φ t ct h 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, Stanford) Systemic Risk March 215 7 / 28

Model Two classes of agents: households and bankers Households:»Z E e ρt 1 1 γ C1 γ t dt, C t = `c y 1 φ φ t ct h 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. TFP shocks dk t K t = i tdt δdt + σdz t He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 7 / 28

Model Two classes of agents: households and bankers Households:»Z E e ρt 1 1 γ C1 γ t dt, C t = `c y 1 φ φ t ct h 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. TFP shocks 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, Stanford) Systemic Risk March 215 7 / 28

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, Stanford) Systemic Risk March 215 8 / 28

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" benchmark capital structure He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 9 / 28

Equity Dynamics in GE Loans to Capital Producers i t Intermediary Sector Household Sector Capital q tk t -1% Housing P th -1% Lev Financial Wealth Equity E t -1% W t = q tk t + P th (1 λ)w t Debt W t E t λw t = "Liquid balances" He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 1 / 28

Equity Constraint Loans to Capital Producers i t Intermediary Sector Aggregate intermediary equity constraint E t de t E t = m ROE, ROE is endogenous 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 = "Liquid balances" He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 11 / 28

Equity constraint: ɛ t Bank can raise equity upto ɛ t at zero cost Cost of raising equity more than ɛ t is infinite. ɛ t linked to intermediary performance (constant m) dɛ t ɛ t = md R t. He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 12 / 28

Equity constraint: ɛ t Bank can raise equity upto ɛ t at zero cost Cost of raising equity more than ɛ t is infinite. ɛ t linked to intermediary performance (constant m) dɛ t ɛ t = md R t. ɛt as reputation" of the banker ɛt as banker s net worth" fluctuating with past returns He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 12 / 28

Equity constraint: ɛ t Bank can raise equity upto ɛ t at zero cost Cost of raising equity more than ɛ t is infinite. ɛ t linked to intermediary performance (constant m) dɛ t ɛ t = md R t. ɛt as reputation" of the banker ɛt as banker s net worth" fluctuating with past returns Aggregate dynamics of E t = R ɛ t He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 12 / 28

Calibration: Baseline Parameters Parameter Choice Targets (Unconditional) Panel A: Intermediation m Performance sensitivity 2 Average Sharpe ratio (model=38%) λ Debt ratio.67 Average intermediary leverage η Banker exit rate 13% Prob. of crisis (model,data = 3%) γ Entry trigger 6.5 Highest Sharpe ratio β Entry cost 2.43 Average land price vol (model,data=14%) Panel B: Technology σ Capital quality shock 3% Consumption volatility (model=1.4%) Note: Model investment vol = 4.5% δ Depreciation rate 1% Literature κ Adjustment cost 3 Literature A Productivity.133 Average investment-to-capital ratio Panel C: Others ρ Time discount rate 2% Literature ξ 1/EIS.15 Interest rate volatility φ Housing share.5 Housing-to-wealth ratio He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 13 / 28

Results(1): State variable is e t = E t /K t 8 Sharpe ratio.1 interest rate 6.5 4 2.5 1.5 5 1 15 2 1 q(e), capital price.95 5 1 15 2 scaled intermediary reputation e.1 5 1 15 2.15.1.95.9 investment I/K.85 5 1 15 2 scaled intermediary reputation e Capital constraint binds for e <.435 He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 14 / 28

Results(2).8 p(e), scaled housing price 1.5 q(e), capital price.6.4 1.2 5 1 15 2 1.8.6.4.2 return volatility of housing 5 1 15 2 scaled intermediary reputation e.95 5 1 15 2.4.3.2.1 steady state distribution 2 4 6 8 scaled intermediary reputation e Capital constraint binds for e <.435 Without the possibility of the capital constraint, all of these lines would be flat. Model dynamics would be i.i.d., with vol=3% He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 15 / 28

State-dependent Impulse Response: -1% Shock (= σdz t ) VARdata.11 Investment.45 Sharpe ratio.1 Land price.12.4.2.13.35.3.14.3.4.15.16.17 crisis normal.25.2.15.1 crisis normal.5.6.7 crisis normal.18.5.8.19 2 4 6 8 quarter 2 4 6 8 quarter.9 2 4 6 8 quarter He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 16 / 28

Matching the 27-29 Crisis Pick initial condition for intermediary state variable (e) to match asset prices in 27Q2 Asset price = Gilchrist-Zakrajsek credit spread Note: this spread (as with most spreads) was low in 27Q2 Data from 1975 to 21; compute histogram of spread variable Match percentile of spread in the data to the same percentile in model implied distribution for risk premium Answer: In 27Q2, e = 1.27. He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 17 / 28

Matching Recent Crisis: Data(L) and Model(R)!"!" Set initial condition of e = 1.27 in 27Q2. Then choose (Z t+1 Z t) shocks to match realized intermediary equity series. 7QIII 7QIV 8QI 8QII 8QIII 8QIV 9QI 9QII 9QIII 9QIV -2.5% -4.2-1.1-1.1 -.7-1.6-1.8-1.8 -.9 -.9 Total -15.5%. Capital constraint binds after 7Q4 systemic risk state In the model (data), land price falls by 5% (55%) In the model (data), investment falls by 23% (25%) He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 18 / 28

Systemic Risk: What is the probability of the 27-29 crisis? What is the likelihood of the constraint binding ( systemic crisis") assuming e = 1.27 currently:.32% in next 1 years 3.57% in next 2 years 17.3% in next 5 years He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 19 / 28

Systemic Risk: What is the probability of the 27-29 crisis? What is the likelihood of the constraint binding ( systemic crisis") assuming e = 1.27 currently: Lessons:.32% in next 1 years 3.57% in next 2 years 17.3% in next 5 years Initial condition calibrated to asset prices + rational forward looking agents = can t see around corners! Even with a highly non-linear model Could abandon RE. Credit growth unusually high, crash likely, even though asset markets dont see it He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 19 / 28

Stress testing: Leverage test Financial sector aggregate leverage fixed at 3 in model We measure across commercial banks, broker/dealers, hedge funds in 27: Assets = $15,73 billion; Liabilities = $1,545 billion Suppose a stress test uncovered leverage: ABCP (SIVs): $1,189 billion; Liabilities $1,189 billion Repo (MMFs and Sec Lenders): $1,2 billion; Liabilities $1, billion (assumed 2% haircut) Leverage is hidden" in sense that agents take equilibrium functions as given based on leverage=3 1 year: 6.73% 2 year: 23.45% 5 year: 57.95 % He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 2 / 28

Stress testing plus a model In current practice, work goes into estimating exposure (i.e. true leverage in example) With a model: 1 Stress may trigger macro and asset price feedbacks, second round,... third round... Model computes the fixed point 2 Model translates stress event into a probability of a systemic crisis 3 Model can help calibrate corrective actions (i.e. capital raising) based on target: How much capital is needed to ensure probability of crisis < X%? Macro-VAR" He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 21 / 28

Stress testing Key step: Need to map from stress scenario into underlying shock, dz t. Say stress scenario -3% Return on equity Naive partial eqbm: leverage of 3, σ(z t+.25 Z t) = 3/3 = 1%. Feed in 1% shock into the model over one quarter. Result: Beginning at e = 1.27 in 27Q2, economy is immediately moved into crisis region, e <.435 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% -1.16% 5.25 % -5-2.53% 8.9-1 -4.69% 22.88-15 -6.71% 48.9-3 -8.72% 1. He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 22 / 28

Stress testing 1 Probability of being distressed: hitting e distress =1.27 1 Probability of capital constraint being binding: hitting e crisis =.4354.9.9.8.8.7.6.5.4.3 in next 2 years in next 5 years in next 1 years.7.6.5.4.3.2 in next 2 years in next 5 years in next 1 years.2.1.1.4.6.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 starting value e init.4.6.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 starting value e init Map stress test" into a shock to e. He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 23 / 28

Conclusion We develop a fully stochastic model of a systemic crisis, with an equity capital constraint on the intermediary sector Is able to replicate 27/28 period with only intermediary capital shocks The model quantitatively matches the differential comovements in distress and non-distress periods Offers a way of mapping macro-stress tests into probability of systemic states. He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 24 / 28

Other crises 1.3 1.2 1.1 1.9.8.7.6 1.8 1.6 1.4 1.2 1.8.6 1988Q3 1988Q4 1989Q1 1989Q2 1989Q3 1989Q4 199Q1 Panel A: Savings and Loan Crisis 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 Panel B: 1998 Hedge Fund Crisis 14.% 12.% 1.% 8.% 6.% 4.% 2.%.% 1.% 9.% 8.% 7.% 6.% 5.% 4.% 3.% 2.% 1.%.% 1.6 1.5 1.4 1.3 1.2 1.1 1.9.8.7.6 1.% 9.% 8.% 7.% 6.% 5.% 4.% 3.% 2.% 1.%.% 21Q1 21Q2 21Q3 21Q4 22Q1 22Q2 22Q3 22Q4 23Q1 Equity-Data Inv-Data Spread-Data Inv-Model Sharpe-Model Prob2-Model (Right axis) He and Krishnamurthy (Chicago, Stanford) Panel C: 22 Systemic Corporate Bond RiskMarket Crisis March 215 25 / 28

Equity series He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 26 / 28

VIX He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 27 / 28

Nonlinearity: VAR in data Panel IR A: Distress Periods.2 Equity to Equity EBP (credit risk premium) to Equity.5 Investment to Equity.15 1.4.1 2 3.3.2.5 4.1 2 4 6 8 5 2 4 6 8 2 4 6 8 Panel B: Non Distress Periods.14.12.1.8.6.4.2 Equity to Equity 2 4 6 8 1 2 3 4 EBP (credit risk premium) to Equity 5 2 4 6 8 Investment to Equity.5.4.3.2.1 2 4 6 8 He and Krishnamurthy (Chicago, Stanford) Systemic Risk March 215 28 / 28