A Macroeconomic Framework for Quantifying Systemic Risk

A Macroeconomic Framework for Quantifying Systemic Risk Zhiguo He, University of Chicago and NBER Arvind Krishnamurthy, Northwestern University and NBER December 2013 He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December 2013 1 / 36

Financial Crisis in the Model 7 Sharpe ratio 6 5 4 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 scaled intermediary reputation e Note: Capital constraint binds for e < 0.435 He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December 2013 2 / 36

Matching Data: Data(L) and Model(R)!"!" Note: The model does poorly on many standard macro calibration targets (e.g., no labor) Model does well in capturing non-linearity in a select set of economic measures... We will have to argue that our metric is a good one He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December 2013 3 / 36

Systemic Risk: What is the probability of the crisis in early 2007? 0 1 2 3 4 2006q1 2007q1 2008q1 2009q1 2010q1 quarter 50000 100000 150000 200000 Equity EBS Equity cutoff He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December 2013 4 / 36

Systemic Risk: What is the probability of the crisis in early 2007? Based on initial condition chosen to match early 2007 asset prices: 1 year: 0.32% 2 year: 3.57% 5 year: 17.30 % Initial condition + rational forward looking agents = cant see around corners! Stress test: Suppose we assume that roughly $2 trillion of shadow banking, with close to 0% capital, was not known to agents 1 year: 6.73% 2 year: 23.45% 5 year: 57.95 % He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December 2013 5 / 36

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 Data Nonlinearity in model and data Match conditional moments of the data, conditioning on negative (i.e., recession) states 3 Quantify systemic risk Systemic risk: the state where financial intermediation is widely disrupted to affect real activities severely In the model, states where capital constraint binds, crisis state What is the ex-ante (e.g., initial conditions of 2007Q2) likelihood of crisis states? (... low) What makes the probability higher? Economics of stress tests (as opposed to accounting of stress tests) He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December 2013 6 / 36

Agents and Technology 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 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. TFP shocks dk t K t = i tdt δdt + σdz t He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December 2013 7 / 36

Agents and Technology 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 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. 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, Northwestern) Systemic Risk December 2013 7 / 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 December 2013 8 / 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" benchmark capital structure He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December 2013 9 / 36

Equity Matters Loans to Capital Producers i t Intermediary Sector Household Sector Capital q tk t Equity E t Housing P th Debt W t E t Separation of ownership and control Banker maximizes E[ROE] m 2 Var[ROE] Financial Wealth W t = q tk t + P th (1 λ)w t λw t = "Liquid balances" benchmark capital structure He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 10 / 36

Single Bank/Banker Choice of Portfolio and Leverage 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 He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 11 / 36

Bank Choice of Portfolio and Leverage 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 ROE: 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 December2013 12 / 36

Bank Choice of Portfolio and Leverage Capital q tk t Housing P th t equity t debt t Properties (k, h) scales with equity (k, h) increasing in E t[d R t r tdt] (k, h) decreasing in Var t[d R 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 ROE: 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]; m parameter He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 13 / 36

General Equilibrium Intermediary Sector Household Sector Capital q tk t Equity E t 2 Financial Wealth Housing P th Debt W 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 = P t H min[e t,(1 λ)w t ] Given 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 (q t) He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 14 / 36

Equity Dynamics in GE Loans to Capital Producers i t Intermediary Sector Household Sector Capital q tk t -10% Housing P th -10% Lev Financial Wealth Equity E t -10% W t = q tk t + P th (1 λ)w t Debt W t E t λw t = "Liquid balances" Banker maximizes E[ROE] m 2 Var[ROE] He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 15 / 36

Equity Constraint Loans to Capital Producers i t Intermediary Sector Aggregate bank reputation 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" Banker maximizes E[ROE] m 2 Var[ROE] He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 16 / 36

Intermediary Reputation 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: Berk-Green, 04; flow-performance relationship, Warther 95; Chevalier-Ellison, 97 Or, ɛt as banker s net worth" fluctuating with performance Kiyotaki-Moore 97, He-Krishnamurthy 12, Brunnermeier-Sannikov 12 Household invests a maximum of ɛ t dollars of equity capital with this banker He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 17 / 36

Intermediary Reputation 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: Berk-Green, 04; flow-performance relationship, Warther 95; Chevalier-Ellison, 97 Or, ɛt as banker s net worth" fluctuating with performance Kiyotaki-Moore 97, He-Krishnamurthy 12, Brunnermeier-Sannikov 12 Household invests 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 Exogenous death rate η. Endogenous entry dψ t > 0 of new bankers in extreme bad states He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 17 / 36

Equity Capital Constraint Representative household with W t, split between bonds (at least) λw t and equity (at most) (1 λ)w t 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 Return on equity = d R t < 10%: equity is levered claim on assets 2 m > 1 in our calibration He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 18 / 36

Calibration: Baseline Parameters Parameter Choice Targets (Unconditional) Panel A: Intermediation m Performance sensitivity 2 Average Sharpe ratio (model=38%) λ Debt ratio 0.67 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 10% Literature κ Adjustment cost 3 Literature A Productivity 0.133 Average investment-to-capital ratio Panel C: Others ρ Time discount rate 2% Literature ξ 1/EIS 0.15 Interest rate volatility φ Housing share 0.5 Housing-to-wealth ratio He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 19 / 36

Results(1): State variable is e t = E t /K t 8 Sharpe ratio 0.1 interest rate 6 0.05 4 0 2 0.05 1.05 0 0 5 10 15 20 1 q(e), capital price 0.95 0 5 10 15 20 scaled intermediary reputation e 0.1 0 5 10 15 20 0.105 0.1 0.095 0.09 investment I/K 0.085 0 5 10 15 20 scaled intermediary reputation e Capital constraint binds for e < 0.435 He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 20 / 36

Results(2) 0.8 p(e), scaled housing price 1.05 q(e), capital price 0.6 0.4 1 0.2 0 0 5 10 15 20 1 0.8 0.6 0.4 0.2 return volatility of housing 0 0 5 10 15 20 scaled intermediary reputation e 0.95 0 5 10 15 20 0.04 0.03 0.02 0.01 steady state distribution 0 0 2 4 6 8 scaled intermediary reputation e Capital constraint binds for e < 0.435 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, Northwestern) Systemic Risk December2013 21 / 36

State-dependent Impulse Response: -1% Shock (= σdz t ) VARdata 0.011 Investment 0.45 Sharpe ratio 0.01 Land price 0.012 0.4 0.02 0.013 0.35 0.03 0.014 0.3 0.04 0.015 0.016 0.017 crisis normal 0.25 0.2 0.15 0.1 crisis normal 0.05 0.06 0.07 crisis normal 0.018 0.05 0.08 0.019 0 2 4 6 8 quarter 0 0 2 4 6 8 quarter 0.09 0 2 4 6 8 quarter He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 22 / 36

Steady State Distribution 0.04 steady state distribution 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 5 10 15 scaled intermediary reputation e He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 23 / 36

Nonlinearities in Model and Data Model: Data: Distress states = worst 33% of realizations of e (e < 1.27) Compute conditional variances, covariances of intermediary equity growth with other key variables Distress states = worst 33% of realizations of (risk premium in) credit spread We use Gilchrist-Zakrajsek (2011) Excess Bond Premium, which we convert to a Sharpe ratio Excess Bond Premium: risk premium of corporate bonds, presumably reflects distress of financial sector Similar results if using NBER recessions Compute conditional variances, covariances of intermediary equity growth with other key variables He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 24 / 36

EBS time series He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 25 / 36

Matching State-Dependent Covariances Distress Non Distress Data Baseline Data Baseline vol (Eq) 31.48% 34.45 17.54 5.4 vol (I) 8.05% 5.30 6.61 4.2 vol (C) 1.71% 3.54 1.28 1.19 vol (LP) 21.24% 21.04 9.79 9.24 vol (EB) 60.14% 74.20 12.72 7.97 cov (Eq, I) 1.31% 1.05 0.07 0.23 cov (Eq, C) 0.25% -0.96 0.03-0.05 cov (Eq, LP) 4.06% 5.87 0.12 0.5 cov (Eq, EB) -6.81% -14.95-0.14-0.13 Note: without the capital constraint, all volatilities would be 3%, and have no state dependence. What we do badly on: Output vol is locally σ because Y t = AK t. Financial friction only affects split between I and C. He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 26 / 36

Matching the 2007-2009 Crisis 0 1 2 3 4 2006q1 2007q1 2008q1 2009q1 2010q1 quarter 50000 100000 150000 200000 Equity EBS Equity cutoff He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 27 / 36

Matching Recent Crisis: Data(L) and Model(R)!"!" Based on EBS classification, economy crossed the 33% boundary (e = 1.27) between 2007Q2 and 2007Q3. Assume e = 1.27 in 2007Q2. Then choose (Z t+1 Z t) shocks to match realized intermediary equity series. 07QIII 07QIV 08QI 08QII 08QIII 08QIV 09QI 09QII 09QIII 09QIV -2.5% -4.2-1.1-1.1-0.7-1.6-1.8-1.8-0.9-0.9 Total -15.5%. Capital constraint binds after 07Q4 systemic risk state In the model (data), land price falls by 50% (55%) In the model (data), investment falls by 23% (25%) He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 28 / 36

Systemic Risk: What is the probability of the 2007-2009 crisis? Small... Based on EBS classification, we cross the 33% boundary (e = 1.27) between 2007Q2 and 2007Q3 What is the likelihood of the constraint binding ( systemic crisis") assuming e = 1.27 currently: 0.32% in next 1 years 3.57% in next 2 years 17.30% in next 5 years He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 29 / 36

Stress testing: Hidden" Leverage Financial sector aggregate leverage fixed at 3 in model We measure across commercial banks, broker/dealers, hedge funds in 2007: Assets = $15,703 billion; Liabilities = $10,545 billion Pushed to crisis boundary after a -7% shock. 3.57% prob. of crisis in next 2 years Hidden leverage: ABCP (SIVs): $1,189 billion; Liabilities $1,189 billion Repo (MMFs and Sec Lenders): $1,020 billion; Liabilities $1,000 billion (assumed 2% haircut) Hidden in sense that agents take as given price functions and returns at leverage=3 1 year: 6.73% 2 year: 23.45% 5 year: 57.95 % He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 30 / 36

Stress testing Key step: Need to map from stress scenario into underlying shock, dz t. Say stress scenario -30% Return on equity Naive partial eqbm: 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 = 1.27 in 2007Q2, economy is immediately moved into crisis region, e < 0.435 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% 8.90-10 -4.69% 22.88-15 -6.71% 48.90-30 -8.72% 100.00 He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 31 / 36

Stress testing 1 Probability of being distressed: hitting e distress =1.27 1 Probability of capital constraint being binding: hitting e crisis =0.4354 0.9 0.9 0.8 0.8 0.7 0.6 0.5 0.4 0.3 in next 2 years in next 5 years in next 10 years 0.7 0.6 0.5 0.4 0.3 0.2 in next 2 years in next 5 years in next 10 years 0.2 0.1 0.1 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 starting value e init 0 0.4 0.6 0.8 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, Northwestern) Systemic Risk December2013 32 / 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 December2013 33 / 36

Equity series He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 34 / 36

VIX He and Krishnamurthy (Chicago, Northwestern) Systemic Risk December2013 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 December2013 36 / 36