A Macroeconomic Model with Financial Panics

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1 A Macroeconomic Model with Financial Panics Mark Gertler, Nobuhiro Kiyotaki, Andrea Prestipino NYU, Princeton, Federal Reserve Board 1 September The views expressed in this paper are those of the authors and do not necessarily reflect the views of the Federal Reserve Board or the Federal Reserve System Gertler Kiyotaki Prestipino September / 31

2 What we do Incorporate banks and banking panics in simple macro model Broad goal: Develop framework to understand dynamics of recent financial crisis Specific goals: Characterize sudden/discrete nature of financial collapse in fall 28 No observable large exogenous shock Gorton (21), Bernanke (21): Bank runs at heart of collapse Model credit boom preceding crisis Optimistic beliefs before crisis (Bordalo et al (217)) Increases susceptibility to runs Gertler Kiyotaki Prestipino September / 31

3 Motivation GDP Growth, Credit Spreads, and Broker Liabilities during the Financial Crisis 1. GDP Growth and Credit Spreads 2. Broker Liabilities Lehman failure 2.5 Lehman failure Nominal GDP Growth -1.5 BAA-1 Year Treasury Spread Gertler Kiyotaki Prestipino September / 31

4 How we differ Conventional financial accelerator/credit cycle models (e.g. Gertler/Kiyotaki 211) Mutual feedback between borrower balance sheets and real activity Local approximations dynamics linear Models with occasionally binding balance sheet constraints (e.g.brunnermeier/sannikov 214, He/Krishnamurthy, 216) Moving from unconstrained to constrained region nonlinear contraction This paper: both occasionally binding constraints and bank runs Runs more significant source of non-linearity Richer macro model Gertler Kiyotaki Prestipino September / 31

5 Model Overview Simple New Keynesian model with investment Banks intermediate funds between households and productive capital Hold imperfectly liquid long term assets and issue short term debt Vulnerable to panic failure of depositors to roll over short term debt Based on GK (215) and GKP (216) In turn based on Cole/Kehoe(21) self-fulfilling sovereign debt Households may directly finance capital, but less efficient at margin than banks Gertler Kiyotaki Prestipino September / 31

6 Evolution and Financing of Capital End of period capital S t vs. beginning K t Γ >, Γ < S t = Γ(I t ) + (1 δ)k t S t K t+1 : ξ t+1 capital quality shock K t+1 = ξ t+1 S t S b t intermediated by banks; S h t directly held by households S t = S b t + S h t Gertler Kiyotaki Prestipino September / 31

7 Household and Bank intermediation Marginal rate of return on intermediated capital R b t+1 = ξ t+1 Z t+1+(1 δ)q t+1 Q t If S h t /S t > γ, (utility) cost to household of direct finance ς(s h t, S t ) = χ 2 ( Sh t S t γ) 2 S t Marginal rate of return on directly held capital with ς( ) S h t R h t+1 = 1 = max 1+ ζ( ) S h t R b t+1 { } χ( Sh t S t γ), For S h t /S t > γ, increasing marginal cost of direct finance Gertler Kiyotaki Prestipino September / 31

8 Household and Bank Intermediation NO BANK RUN EQUILIBRIUM Q t S b t D t CAPITAL S t N t! " #$ % h Q t S t HOUSEHOLDS CAPITAL S t BANK RUN EQUILIBRIUM Q* t S t HOUSEHOLDS Gertler Kiyotaki Prestipino September / 31

9 Bank Decision Problem Objective V t = E t Λ t,t+1 [(1 σ)n t+1 + σv t+1 ] σ exogenous survival probability Net worth n t accumulated via retained earnings - no new equity issues n t+1 = R k t+1q t s b t R t+1 d t if no run = if run Balance sheet Q t s b t = d t + n t Gertler Kiyotaki Prestipino September / 31

10 Deposit Contract R t+1 deposit rate; R t+1 return on deposits p t run probability; x t+1 < 1 recovery rate Deposit contract: (One period) { Rt+1 with prob. 1 p R t+1 = t x t+1 R t+1 with prob. p t Recovery rate (no sequential service constraint): x t+1 = ξ [ t+1 Zt+1 + (1 δ) Qt+1 ] S b t R t+1 D t Gertler Kiyotaki Prestipino September / 31

11 Bank Decision Problem: Perfect vs. Imperfect Markets Perfect markets: Banks issue deposits until: E t Λ t,t+1 {R k t+1 R t+1 } = Leverage constraints do not arise Financial panics cannot arise Limits to arbritage: Occasionally binding leverage constraints E t Λ t,t+1 {Rt+1 k R t+1 } > Bank runs possible: extreme increases in E t Λ t,t+1 {Rt+1 k R t+1} Gertler Kiyotaki Prestipino September / 31

12 Limits to Bank Arbitrage Moral Hazard Problem: After banker borrows funds at t, it may divert fraction θ of assets for personal use. If bank does not honor its debt, creditors can recover the residual funds and shut the bank down. Incentive constraint (IC) θq t s b t V t Gertler Kiyotaki Prestipino September / 31

13 Solution Can show V t = ψ t n t with ψ t 1 and increasing in E t {R k t+1 R t+1} Combine with IC endogenous leverage constraint : Q t s b t φ t n t φ t = ψt θ decreasing in θ and increasing in E t{r k t+1 R t+1} Note: E t {R k t+1 R t+1} countercyclical φ t countercyclical. n t bank cannot operate (key for run equilbria) Gertler Kiyotaki Prestipino September / 31

14 Aggregation: No Run Case Homogeneity: φ t Qtsb t n t and φ t independent of bank-specific factors Aggregate leverage constraint Q t S b t φ t N t Aggregate net worth E t Λ t,t+1 {R k t+1 R t+1 } > N t = σ[(r k t R t )φ t 1 + R t ]N t 1 + ζs t 1 Absent runs, conventional financial accelerator with non-linearity Gertler Kiyotaki Prestipino September / 31

15 Bank Runs Self-fulfilling bank run equilibrium (i.e. rollover crisis) possible if: A depositor believes that if other households do not roll over their deposits, the depositor will lose money by rolling over. Condition met if banks net worth n t goes to zero if others run n t = banks cannot operate Run equilibrium exists if recovery rate x t satisfies x t = ξ t(z t + (1 δ)q t )S b t 1 R t D t 1 < 1 x t < 1 n t = after run Gertler Kiyotaki Prestipino September / 31

16 Run and Run Probability p t Run equilibrium occurs if Run equilibrium exists Sunspot is observed Assume sunspot occurs with probability κ. The time t probability of a run at t + 1 is p t = Pr t {x t+1 < 1} κ Pr t {x t+1 < 1} countercyclical p t countercyclical Gertler Kiyotaki Prestipino September / 31

17 Liquidation Price Q t After bank run at t : S h t = S t Household euler equation for capital Q t = E t {(Λ t,t+1 ξ t+1 (Z t+1 + (1 δ)q t+1 )} χ( S h t S t γ) 1 λ t evaluated at Sh t S t = 1. Q t < Q t At t + 1 new banks enter and assets slowly return to banking system Gertler Kiyotaki Prestipino September / 31

18 Production, Pricing and Monetary Policy (Standard) Production, resource constraint and Q relation for investment Y t = AK α t L 1 α t Y t = C t + I t + G Q t = Φ(I t ) Monopolistically comp. producers with quadratic costs of nominal price adjustment (Rotemberg) Adjust output to meet demand New Keynesian Phillips curve relating inflation to marginal cost Monetary policy: simple Taylor rule R n t = 1 β ( P t P t 1 ) κπ (Θ t ) κy Gertler Kiyotaki Prestipino September / 31

19 Calibration Parameter Description Value Target Standard Parameters β Impatience.99 Risk Free Rate γ h Risk Aversion 2 Literature ϕ Frish Elasticity 2 Literature ɛ Elasticity of subst across varieties 11 Markup 1% α Capital Share.33 Capital Share δ Depreciation.25 I K =.25 η Elasticity of q to i.25 Literature a Investment Technology Parameter.53 Q = 1 b Investment Technology Parameter -.83% I K =.25 G G Government Expenditure.45 Y =.2 ρ jr Price adj costs 1 Slope of Phillips curve.1 κ π Policy Response to Inflation 1.5 Literature κ y Policy Response to Output.5 Literature Financial Intermediation Parameters σ Banker Survival rate.93 Leverage QSb N = 1 New Bankers Endowments ζ.1% % I in crisis 35% as a share of Capital θ Share of assets divertible.22 Spread Increase in Crisis = 1.5% γ Threshold for S.61 b HH Intermediation Costs S =.33 χ HH Intermediation Costs.15 ER b R = 2% Annual κ Sunspot Probability.15 Run Probability 4% Annual σ(ɛ ξ ) std of innovation to capital quality.5% std Output (C+I) ρ ξ serial correlation of capital quality.7 std Investment Gertler Kiyotaki Prestipino September / 31

20 Level Annual Basis Points Level Annual Basis Points " from SS Annual basis points Level Response to Response a Capital to a Capital Quality Shock: (1 std): No Run Case Case Baseline No Financial Fricitons Capital Quality.1 Run Probability 5 Bank Net Worth -.5 Capital Quality Run Threshold Leverage Multiple:? Investment Output Excess Return: ER b -R free 45 Policy Rate 5 Inflation Quarters Quarters Quarters Gertler Kiyotaki Prestipino September / 31

21 Level Annual Basis Points Level Annual Basis Points Level Annual Basis Points Level Response to a Sequence of Shocks: Run VS No Run Response to a Sequence of Shocks: Run VS No Run RUN (Run Threshold Shock and Sunspot) NO RUN (Run Threshold Shock and No Sunspot) Capital Quality Capital Quality Run Threshold Initial Threshold.15.1 Run Probability -5 Bank Net Worth Leverage Multiple:? Investment Output Excess Return: ER b -R free 6 Policy Rate 1 Inflation Quarters Quarters Quarters Gertler Kiyotaki Prestipino September / 31

22 Level Annual Basis Points Level Annual Basis Points Level Response to a Sequence of Shocks in Flex Price Economy: Run VS No Run Response to the Same Sequence of Shocks in Flex Price Economy: Run VS No Run RUN (Run Threshold Shock and Sunspot) NO RUN (Run Threshold Shock and No Sunspot) Capital Quality.8 Run Probability Bank Net Worth -2 Capital Quality Run Threshold Leverage:? Investment Output Excess Return: ER B -R free 5 Natural Rate 1 Consumption Quarters Quarters Quarters Gertler Kiyotaki Prestipino September / 31

23 Financial Crisis: Model vs Data Shocks : Threshold : Financial Crisis: Model vs. Data -.2 % -.5 % -.4 % -.6 % -.6 % -.9 % -.8 % -.7 % -.7 % -.6 % 27q4 28q1 28q2 28q3 28q4 1. Investment 2. XLF Index and Net Worth 1 25 Bear Stearns Lehman Brothers 3. Spreads (AAA-Risk Free) 3 Data Model Model No Run q3 28q q3 28q q3 28q GDP 5 5. Labor (hours) 5 6. Consumption q3 28q q3 28q q3 28q4 216 NOTE: The data for GDP, Investment, and Consumption are computed as logged deviations from trend where the trend is the CBO potential GDP. Labor data is computed as logged deviations from trend where the trend is the CBO potential hours worked. The XLF Index data is computed as the percent deviation from its 27q3 level. Gertler Kiyotaki Prestipino September / 31

24 Boom leading to the bust: news driven optimism Capital quality: ξ t+1 = ρ ξ ξ t + ɛ ξ t+1 At t = bankers learn that unusually large realization of ɛ ξ t+1 B > will happen at t B {1,..., T } with prob. P B < 1 of size Pr {t B = t} is a truncated Normal (discrete approx.) Agents update Pr t and P B t by observing ɛ ξ t Prob. at t of shock at t + 1 is Pr t {t B = t + 1} P B t Implies forecast errors in line with evidence, e.g. Bordalo et al 217 Gertler Kiyotaki Prestipino September / 31

25 Level Optimism, credit boom and financial vulnerability (no run) Prior cond. prob. of shock happening at time t.2 1 Beliefs Evolution.6 Capital Quality P B t Pr tft b = t + 1g t E t 9 t Q1 27 Q2 Time 28 Q4 25 Q1 27 Q2 28 Q4 Time 25 Q1 27 Q2 28 Q4 1 Output 12 Debt.3 Probability of being in crisis zone Q1 27 Q2 28 Q Q1 27 Q2 28 Q4 25 Q1 27 Q2 28 Q4 Gertler Kiyotaki Prestipino September / 31

26 Financial Crisis After Credit Boom: Model vs Data Financial Crisis after Credit Boom: Model vs. Data Shocks : -.2 % -.4 % -.3 % -.5 % -. % Threshold : -.1 % -.1 %. % -. % -. % 27q4 28q1 28q2 28q3 28q4 1. Investment 2. XLF Index and Net Worth 3. Spreads (AAA-Risk Free) 1 25 Bear Stearns Lehman Brothers 3 Data Model Model No Run q3 28q q3 28q q3 28q GDP 5 5. Labor (hours) 5 6. Consumption q3 28q q3 28q q3 28q4 216 potential GDP. Labor data is computed NOTE: The data for GDP, Investment, and Consumption are computed as logged deviations from trend where the trend is the CBO where Gertler the trend Kiyotaki is the CBO potential Prestipino hours worked. The XLF Index data is computed as the percent deviation from its 27q3 level. as logged deviations from trend September / 31

27 Forecast Errors in Credit Spreads (Baa-1yr Treasury) Financial Crisis: Forecast Error Forecast Errors: AAA-Treasury (4-Quarters Ahead) Bear Stearns Lehman Brothers Data (Bordalo-Gennaioli-Shleifer) Model Error (Next 4Q Average) = Actual - Forecast q3 27q3 28q4 213q4 Gertler Kiyotaki Prestipino September / 31-1

28 Conclusion Incorporated banking sector with conventional macro model Banks occasionally exposed to self-fulfilling rollover crises Crises lead to significant contractions in real economic activity Model captures qualitatively and quantitatively Nonlinear dimension of financial crises The broad features of the recent recent collapse Credit boom preceding crisis Next steps: Macroprudential policy (Run Externality) Lender-of-last resort policies Gertler Kiyotaki Prestipino September / 31

29 Run Equilibrium Threshold No Run-Equilibrium Possible A Negative Capital Quality Shock B Run-Equilibrium Possible 1 Gertler Kiyotaki Prestipino September / 31

30 Conditions for Bank Run Equilibrium with We can simplify existence condition for BRE: x t = Rb t φ t 1 R t φ t 1 1 < 1 R b t = ξt[zt+(1 δ)q t ] Q t 1 ; φ t 1 = Q t 1S b t 1 N t 1 Likelihood BRE exists decreasing in Q ( ) and increasing in φ t 1 φ t 1 countercyclical likelihood BRE exists is countercyclical. Gertler Kiyotaki Prestipino September / 31

31 Run Equilibrium Threshold RUN THRESHOLD No Run-Equilibrium Possible A.8 Negative Capital Quality shock.6 B Run-Equilibrium Possible Gertler Kiyotaki Prestipino September / 31

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