Banking Crises as Self-Defeating Prophecies

Transcription

1 Banking Crises as Self-Defeating Prophecies Yaniv Ben-Ami Yale University February 22, 2017 Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

2 Overview Bottom Line In misspecified models projected prices do not equal realizations. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

3 Overview Bottom Line In misspecified models projected prices do not equal realizations. But these realizations also do not equal the rational expectations prices. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

4 Overview Bottom Line In misspecified models projected prices do not equal realizations. But these realizations also do not equal the rational expectations prices. A self-defeating prophecy: when a prediction mistake pushes the realization away from the rational expectations opposite the direction of the prediction. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

5 Overview Bottom Line In misspecified models projected prices do not equal realizations. But these realizations also do not equal the rational expectations prices. A self-defeating prophecy: when a prediction mistake pushes the realization away from the rational expectations opposite the direction of the prediction. Can banking crises be partially due to self-defeating prophecies? Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

6 Overview Bottom Line In misspecified models projected prices do not equal realizations. But these realizations also do not equal the rational expectations prices. A self-defeating prophecy: when a prediction mistake pushes the realization away from the rational expectations opposite the direction of the prediction. Can banking crises be partially due to self-defeating prophecies? I calibrate a novel banking model and find 3-fold prediction error amplification. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

7 Overview Bottom Line In misspecified models projected prices do not equal realizations. But these realizations also do not equal the rational expectations prices. A self-defeating prophecy: when a prediction mistake pushes the realization away from the rational expectations opposite the direction of the prediction. Can banking crises be partially due to self-defeating prophecies? I calibrate a novel banking model and find 3-fold prediction error amplification. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

8 Overview The Classic Example (Simplistic) Jonah the failed prophet. Then God saw their works, that they turned from their evil way; and God relented from the disaster that He had said He would bring upon them, and He did not do it. But it displeased Jonah exceedingly, and he became angry. Jonah 3:10-4:1, NKJV Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

9 Overview Preview: Self-Defeating Prophecies: Phase I Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

10 Overview Model Overview Exogenous non-tradable stochastic growth endowment. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

11 Overview Model Overview Exogenous non-tradable stochastic growth endowment. Collateral asset is real-estate, in fixed supply. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

12 Overview Model Overview Exogenous non-tradable stochastic growth endowment. Collateral asset is real-estate, in fixed supply. Rent follows the non-tradable endowment 1-for-1. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

13 Overview Model Overview Exogenous non-tradable stochastic growth endowment. Collateral asset is real-estate, in fixed supply. Rent follows the non-tradable endowment 1-for-1. Contracts are collateralized debt obligations - all leverage levels are available - no penalty for default. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

14 Overview Model Overview Exogenous non-tradable stochastic growth endowment. Collateral asset is real-estate, in fixed supply. Rent follows the non-tradable endowment 1-for-1. Contracts are collateralized debt obligations - all leverage levels are available - no penalty for default. Three types of agents - only differ by risk aversion. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

15 Overview Model Overview Exogenous non-tradable stochastic growth endowment. Collateral asset is real-estate, in fixed supply. Rent follows the non-tradable endowment 1-for-1. Contracts are collateralized debt obligations - all leverage levels are available - no penalty for default. Three types of agents - only differ by risk aversion. Risk aversion heterogeneity drives lending and borrowing. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

16 Overview Model Overview Exogenous non-tradable stochastic growth endowment. Collateral asset is real-estate, in fixed supply. Rent follows the non-tradable endowment 1-for-1. Contracts are collateralized debt obligations - all leverage levels are available - no penalty for default. Three types of agents - only differ by risk aversion. Risk aversion heterogeneity drives lending and borrowing. Intermediate risk aversion type emerges as a bank. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

17 Overview Model Overview Exogenous non-tradable stochastic growth endowment. Collateral asset is real-estate, in fixed supply. Rent follows the non-tradable endowment 1-for-1. Contracts are collateralized debt obligations - all leverage levels are available - no penalty for default. Three types of agents - only differ by risk aversion. Risk aversion heterogeneity drives lending and borrowing. Intermediate risk aversion type emerges as a bank. Endowment process is calibrated to consumption disasters data. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

18 Overview Model Overview Exogenous non-tradable stochastic growth endowment. Collateral asset is real-estate, in fixed supply. Rent follows the non-tradable endowment 1-for-1. Contracts are collateralized debt obligations - all leverage levels are available - no penalty for default. Three types of agents - only differ by risk aversion. Risk aversion heterogeneity drives lending and borrowing. Intermediate risk aversion type emerges as a bank. Endowment process is calibrated to consumption disasters data. Sensitivity of asset pricing to wealth distribution is calibrated to match U.S. cyclical variation in risk-return ratio. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

19 Overview High Leverage vs. High Risk Premium The model has to account for realistically high leverage, while also matching a high risk premium. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

20 Overview High Leverage vs. High Risk Premium The model has to account for realistically high leverage, while also matching a high risk premium. Availability of high Loan-To-Value (LTV) indicates agents do not foresee large price drops. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

21 Overview High Leverage vs. High Risk Premium The model has to account for realistically high leverage, while also matching a high risk premium. Availability of high Loan-To-Value (LTV) indicates agents do not foresee large price drops. A high risk premium suggests agents are afraid to hold assets. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

22 Overview High Leverage vs. High Risk Premium The model has to account for realistically high leverage, while also matching a high risk premium. Availability of high Loan-To-Value (LTV) indicates agents do not foresee large price drops. A high risk premium suggests agents are afraid to hold assets. I build on the disaster-based asset pricing literature. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

23 Overview High Leverage vs. High Risk Premium The model has to account for realistically high leverage, while also matching a high risk premium. Availability of high Loan-To-Value (LTV) indicates agents do not foresee large price drops. A high risk premium suggests agents are afraid to hold assets. I build on the disaster-based asset pricing literature. Instead of disasters that are immediately recognizable to agents, I make disasters unfold from regular recessions. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

24 Overview High Leverage vs. High Risk Premium The model has to account for realistically high leverage, while also matching a high risk premium. Availability of high Loan-To-Value (LTV) indicates agents do not foresee large price drops. A high risk premium suggests agents are afraid to hold assets. I build on the disaster-based asset pricing literature. Instead of disasters that are immediately recognizable to agents, I make disasters unfold from regular recessions. Because price declines are gradual, LTV is high. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

25 Overview High Leverage vs. High Risk Premium The model has to account for realistically high leverage, while also matching a high risk premium. Availability of high Loan-To-Value (LTV) indicates agents do not foresee large price drops. A high risk premium suggests agents are afraid to hold assets. I build on the disaster-based asset pricing literature. Instead of disasters that are immediately recognizable to agents, I make disasters unfold from regular recessions. Because price declines are gradual, LTV is high. The risk premium is high because of the fear of financial dysfunction. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

26 Overview Tree Structure Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

27 Overview Result Preview: Sharpe Ratio Dynamics s w(s) p h (s) SR(s) H r (s) H b (s) H d (s) % 34% 0% u % 10% 0% d % 70% 0% uu % 0% 0% ud % 46% 0% du % 37% 0% dd % 57% 15% Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

28 Overview Result Preview: Prediction Error Amplification (PEA) Predicted Realized p h (d) PEA(p h (d), g d ) g d g d Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

29 Overview PEA s Empirical Content PEA can be empirically measured (in theory). Measurement requires: 1 The model agents used to make decisions. 2 The errors they made. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

30 Overview PEA s Empirical Content PEA can be empirically measured (in theory). Measurement requires: 1 The model agents used to make decisions. 2 The errors they made. Example: the subprime mortgage crisis. 1 The model: Collateralized Debt Obligations credit rating models. 2 The errors: Too low correlations assumed across geographic regions. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

31 Overview Related Literature References This paper extends Nakamura, Steinsson, Barro, and Ursúa (2013) to account for leverage and Sharpe ratio dynamics. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

32 Overview Related Literature References This paper extends Nakamura, Steinsson, Barro, and Ursúa (2013) to account for leverage and Sharpe ratio dynamics. It s an application of Fostel and Geanakoplos (2008) and Fostel and Geanakoplos (2012) to banking, where risk aversion heterogeneity replaces belief and endowment heterogeneity. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

33 Overview Related Literature References This paper extends Nakamura, Steinsson, Barro, and Ursúa (2013) to account for leverage and Sharpe ratio dynamics. It s an application of Fostel and Geanakoplos (2008) and Fostel and Geanakoplos (2012) to banking, where risk aversion heterogeneity replaces belief and endowment heterogeneity. It s a response to the view that banking crises are the result of equilibrium multiplicity, as for example in Gertler and Kiyotaki (2015). Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

34 Overview Related Literature References This paper extends Nakamura, Steinsson, Barro, and Ursúa (2013) to account for leverage and Sharpe ratio dynamics. It s an application of Fostel and Geanakoplos (2008) and Fostel and Geanakoplos (2012) to banking, where risk aversion heterogeneity replaces belief and endowment heterogeneity. It s a response to the view that banking crises are the result of equilibrium multiplicity, as for example in Gertler and Kiyotaki (2015). Similar to Greenwood, Hanson, and Jin (2016) and Bordalo, Gennaioli, and Shleifer (2016) it studies credit cycles under mistakes, but I do not commit to systemic mistakes... Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

35 Overview Related Literature References This paper extends Nakamura, Steinsson, Barro, and Ursúa (2013) to account for leverage and Sharpe ratio dynamics. It s an application of Fostel and Geanakoplos (2008) and Fostel and Geanakoplos (2012) to banking, where risk aversion heterogeneity replaces belief and endowment heterogeneity. It s a response to the view that banking crises are the result of equilibrium multiplicity, as for example in Gertler and Kiyotaki (2015). Similar to Greenwood, Hanson, and Jin (2016) and Bordalo, Gennaioli, and Shleifer (2016) it studies credit cycles under mistakes, but I do not commit to systemic mistakes and they used mis-perception of probabilities, which cannot result in erroneous conditional predictions. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

36 Model Table of Contents 1 Overview 2 Model 3 Calibration 4 Results under Rational Expectations 5 Results around Rational Expectations 6 Discussion Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

37 Model Notation Time is discrete and infinite and the state of the world is a binary tree. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

38 Model Notation Time is discrete and infinite and the state of the world is a binary tree. The set of all states of the world is denoted S. The root of the tree is denoted 0. Each other node is denoted by the string of moves {u, d} that reaches it from 0. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

39 Model Notation Time is discrete and infinite and the state of the world is a binary tree. The set of all states of the world is denoted S. The root of the tree is denoted 0. Each other node is denoted by the string of moves {u, d} that reaches it from 0. Previous state { 0, s = u or s = d s 1 s, s = s u or s = s d. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

40 Model Notation Time is discrete and infinite and the state of the world is a binary tree. The set of all states of the world is denoted S. The root of the tree is denoted 0. Each other node is denoted by the string of moves {u, d} that reaches it from 0. Previous state { 0, s = u or s = d s 1 s, s = s u or s = s d. Next state set s +1 { {u, d}, s = 0 {su, sd}, s S \ 0. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

41 Model Notation Time is discrete and infinite and the state of the world is a binary tree. The set of all states of the world is denoted S. The root of the tree is denoted 0. Each other node is denoted by the string of moves {u, d} that reaches it from 0. Previous state { 0, s = u or s = d s 1 s, s = s u or s = s d. Next state set s +1 { {u, d}, s = 0 {su, sd}, s S \ 0. Time index { 0, s = 0 t(s) t(s 1 ) + 1, s S \ 0. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

42 Model The Endowment Process w(s) is the exogenous non-tradable endowment. Growth parameters: Pr(sd s) = γ { g d < 0, t(s) < T ln w(sd) ln w(su) = 0, t(s) T E s s +1 ln w(s ) ln w(s) = g Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

43 Model The Endowment Process w(s) is the exogenous non-tradable endowment. Growth parameters: Pr(sd s) = γ { g d < 0, t(s) < T ln w(sd) ln w(su) = 0, t(s) T E s s +1 ln w(s ) ln w(s) = g Growth process: ln w(s 1 ) + g γg d + g d, s = s 1 d and t(s) T, ln w(s) = ln w(s 1 ) + g γg d, s = s 1 u and t(s) T ln w(s 1 ) + g, t(s) > T Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

44 Model Utility Three agent types: r - real estate speculators, b - bankers, d - depositors. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

45 Model Utility Three agent types: r - real estate speculators, b - bankers, d - depositors. Epstein-Zin preferences only differ by the parameter of relative risk aversion Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

46 Model Utility Three agent types: r - real estate speculators, b - bankers, d - depositors. Epstein-Zin preferences only differ by the parameter of relative risk aversion i {r, b, d} : U i (s) [ (1 β)u i (s) 1 η + β ( E s s +1 U i (s ) 1 α i ) 1 η 1 α i ] 1 1 η, α d > α b > α r, Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

47 Model Utility Three agent types: r - real estate speculators, b - bankers, d - depositors. Epstein-Zin preferences only differ by the parameter of relative risk aversion i {r, b, d} : U i (s) [ (1 β)u i (s) 1 η + β ( E s s +1 U i (s ) 1 α i ) 1 η 1 α i ] 1 1 η, α d > α b > α r, where the per-period utility for all agents is a Cobb-Douglas in consumption, c i (s), and rented housing, h i (s), u i (s) c i (s) 1 ρ h i (s) ρ. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

48 Model Budget Constraints House are rented at the beginning of the period and are bought and sold at the end of the period. Payments are made last. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

49 Model Budget Constraints House are rented at the beginning of the period and are bought and sold at the end of the period. Payments are made last. The budget constraint for all agents is c i (s) + p h (s)h i (s) + R(s)h i (s) W i w(s) + (p h (s) + R(s))H i (s 1 ) + B i (s) (1 + r(s 1 ))B i (s 1 ), where c i (s) - consumption, p h (s) - price of real estate, H i (s) - stock of real estate owned at the end of the period, R(s) - rental rate on real estate, h i (s) - the size of the house rented, W i - non-tradable endowment share, w(s) - aggregate labor endowment, B i (s) - amount borrowed, r(s) - interest rate. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

50 Model Collateral Constraints When contracts are limited to collateralized debt and the state space is a binomial tree, risky debt is never needed (traded only in ways that do not affect risk sharing). Proof Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

51 Model Collateral Constraints When contracts are limited to collateralized debt and the state space is a binomial tree, risky debt is never needed (traded only in ways that do not affect risk sharing). Proof Intuitively, this is because the interest demanded excess the willingness to pay. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

52 Model Collateral Constraints When contracts are limited to collateralized debt and the state space is a binomial tree, risky debt is never needed (traded only in ways that do not affect risk sharing). Proof Intuitively, this is because the interest demanded excess the willingness to pay. The implied collateral constraint is (1 + r(s))b i (s) (p h (sd) + R(sd))H i (s) s S. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

53 Model Rental and Goods First Order Conditions The first order conditions on consumption and renting imply R(s)h i (s) ρ = c i(s) 1 ρ i {r, b, d} Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

54 Model Rental and Goods First Order Conditions The first order conditions on consumption and renting imply R(s)h i (s) ρ = c i(s) 1 ρ i {r, b, d} Summing over agents we have R(s) = ρ c(s) 1 ρ h(s) = ρ 1 ρ w(s). Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

55 Model Housing and Debt First Order Conditions Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

56 Model Housing and Debt First Order Conditions Agents first order conditions for housing owned is p h (s) = E s s +1 [ Mi (s ) ( p h (s ) + R(s ) )] + µ i (s)(p h (sd) + R(sd)), Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

57 Model Housing and Debt First Order Conditions Agents first order conditions for housing owned is p h (s) = E s s +1 [ Mi (s ) ( p h (s ) + R(s ) )] + µ i (s)(p h (sd) + R(sd)), Their first order conditions for debt is 1 = (1 + r i (s)) ( E s s +1 [ Mi (s ) ] + µ i (s) ). Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

58 Model Housing and Debt First Order Conditions Agents first order conditions for housing owned is p h (s) = E s s +1 [ Mi (s ) ( p h (s ) + R(s ) )] + µ i (s)(p h (sd) + R(sd)), Their first order conditions for debt is 1 = (1 + r i (s)) ( E s s +1 [ Mi (s ) ] + µ i (s) ). Where the stochastic discount factor is M i (s) = βu i (s) α i λ i (s) λ i (s 1 ) U i(s 1 ) η ( E s s 1+1 [ Ui (s ) 1 α i ]) αi η 1 α i, and the multiplier on the agents budget constraint is Pr(s)λ i (s), and the multiplier on the collateral constraint is Pr(s)λ i (s)µ i (s). Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

59 Model Optional Model Slides Market Clearing Equilibrium Definition Leverage Definitions Asset Pricing Measures Definitions Off Rational Expectations Payments Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

60 Calibration Table of Contents 1 Overview 2 Model 3 Calibration 4 Results under Rational Expectations 5 Results around Rational Expectations 6 Discussion Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

61 Calibration Parameters Calibrated Directly from Data Parameter Value Role Target Source T 3 # of periods with T=4 cannot Assumed. growth uncertainty. be observed in the data. γ 18.5% Probability of a negative (1 γ)γ 2 = Nakamura et al. shock. 2.8% (2013). g d -6.5% Growth loss on each (2 + γ)g d = Nakamura et al. negative shock. 14% (2013). g 2.2% Baseline labor endowment 2.2% Nakamura et al. growth rate. (2013). ρ 17% Share of rent in expen- 17% NIPA Table ditures. η 0.1 Controls the elasticity of intertemporal substitution β Time discount, controls the price to rent under no uncertainty. Small interest rate responses line 15/ line 1 Assumed. p h R = 16.7 Z.1 Table B.1 line 3 + line 8 / NIPA Table line 1 Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

62 Calibration Parameters Calibrated using the Equilibrium Parameter Value Role Target Source α d 50 Depositor relative risk aversion. SR(dd) > 1 Assumed. α s 0.1 Speculator relative EC r (d) 0 Assumed. risk aversion. α b 3 Banker relative risk SR(0) = 0.38 Lustig and Verdelhan aversion. (2012). W d 0.96 Depositor nontradable H d (d) = 0 < Assumed. share. H d (dd) 1 W b, W r 0.02 Banker and speculator 2 d) Assumed. non-tradable share. H b (0 1), Assumed. H r (0 1) 0.04 Banker and speculator initial housing endowment share. H d (0 1) 0.92 Depositor initial housing endowment share. H r (0 1) = 2W r (0 1), H b (0 1) = 2W b (0 1) 1 H r (0 1) H b (0 1) Assumed. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

63 Results under Rational Expectations Table of Contents 1 Overview 2 Model 3 Calibration 4 Results under Rational Expectations 5 Results around Rational Expectations 6 Discussion Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

64 Results under Rational Expectations Sharpe Ratio Dynamics s w(s) p h (s) SR(s) H r (s) H b (s) LEV r (s) LEV b (s) LEV b+r (s) % 34% u % 10% d % 70% uu % 0% ud % 46% du % 37% dd % 57% collateral constrained. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

65 Results under Rational Expectations Risk Premium - Speculator Pricing p h (s) = E s s +1 [M i (s ) (p h (s ) + R(s ))] + µ i (s)(p h (sd) + R(sd)), M i (s) = βu i (s) α i λ i (s) λ i (s 1 ) U i(s 1 ) η ( [ E s s 1+1 Ui (s ) 1 α ]) αi η i 1 α i s RP(s) U r (s) αr λ r (s) U r (s) λ r (s) COH r (s) µ r (s) % u 0.27% d 3.66% uu 0.06% ud 0.95% du 0.68% dd 3.45% Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

66 Results under Rational Expectations Risk Premium - Banker Pricing p h (s) = E s s +1 [M i (s ) (p h (s ) + R(s ))] + µ i (s)(p h (sd) + R(sd)), M i (s) = βu i (s) α i λ i (s) λ i (s 1 ) U i(s 1 ) η ( [ E s s 1+1 Ui (s ) 1 α ]) αi η i 1 α i s RP(s) U b (s) α b λ b (s) U b (s) λ b (s) COH b (s) µ b (s) % u 0.27% d 3.66% uu 0.06% ud 0.95% du 0.68% dd 3.45% Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

67 Results around Rational Expectations Table of Contents 1 Overview 2 Model 3 Calibration 4 Results under Rational Expectations 5 Results around Rational Expectations 6 Discussion Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

68 Results around Rational Expectations Out of Rational Expectations Analysis The model provides a mapping f between different beliefs about g d and agent s portfolio choices: f (g d ) = (H i(0(g d )), EC i(0(g d )), LTV (0(g d )), r(0(g d )) i {r,b,d}. Let s = d(g d! ) be the root node of an equilibrium with T = 2, a shared belief that g d is at its true value, a non-tradable endowment that equals the true w(d), but with initial asset endowments set to (H i (d(g! d ) 1), EC i (d(g! d ) 1), LTV (d(g! d ) 1), r(d(g! d ) 1)) i {r,b,d} = f (g d ). Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

69 Results around Rational Expectations Prediction Error Amplification Prediction Error Amplification is defined as where x is the rational prediction, x is the erroneous prediction, x! is the realization. PEA(x) = x x! x x Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

70 Results around Rational Expectations Prediction Error Amplification Prediction Error Amplification is defined as where x is the rational prediction, x is the erroneous prediction, x! is the realization. PEA(x) = x x! x x When PEA > 1 the environment is hostile in that it makes agents seems less informed than they really are. Prophecies are self-defeating. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

71 Results around Rational Expectations Prediction Error Amplification Prediction Error Amplification is defined as where x is the rational prediction, x is the erroneous prediction, x! is the realization. PEA(x) = x x! x x When PEA > 1 the environment is hostile in that it makes agents seems less informed than they really are. Prophecies are self-defeating. When PEA [0, 1) the environment is friendly in that it makes agents seems more informed than they really are. Prophecies are self-fulfilling. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

72 Results around Rational Expectations Prediction Error Amplification (PEA) Predicted Realized full learning Realized no learning p h (d) PEA(p h (d), g d ) g d g d Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

73 Results around Rational Expectations Self-Defeating Prophecies: Phase I Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

74 Results around Rational Expectations Self-Defeating Prophecies: Phase II Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

75 Results around Rational Expectations Self-Defeating Prophecies: Phase III Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

76 Results around Rational Expectations PEA is a Result of Ownership Shift H d (d) p h (d) Predicted Realized g d g d SR(d) H b (d) g d g d r(d) H r (d) g d g d Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

77 Results around Rational Expectations Ownership Shift is Caused by a Leverage Shift at Root LEV b (0) g d LEV r (0) g d LEV b + r (0) g d H d (0) g d H b (0) g d H r (0) g d Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

78 Results around Rational Expectations Leverage Shift at Root Drives Wealth at Down COH d (d) g d COH b (d) g d COH r (d) g d Losses on mortgages that were perceived as safe visible as a flattening of the COH r (d) curve. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

79 Results around Rational Expectations Leverage at Down Must Increase EC d (d) LEV b (d) g d g d EC b (d) LEV r (d) g d g d EC r (d) g d LEV b + r (d) g d Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

80 Results around Rational Expectations Wealth Effects are Kept Small Wealth effects in s = 0 are a distraction. They are kept small by assuming initial asset endowments from are such that no trade is needed in s = e COH d (0) COH b (0) COH r (0) g d g d g d p h (0) g d SR(0) g d r(0) g d Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

81 Discussion Table of Contents 1 Overview 2 Model 3 Calibration 4 Results under Rational Expectations 5 Results around Rational Expectations 6 Discussion Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

82 Discussion PEA vs. Multiple Equilibria PEA is a more subtle notion of the relation between beliefs and reality than multiple equilibria. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

83 Discussion PEA vs. Multiple Equilibria PEA is a more subtle notion of the relation between beliefs and reality than multiple equilibria. This is because PEA separates belief and the realization with time. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

84 Discussion PEA vs. Multiple Equilibria PEA is a more subtle notion of the relation between beliefs and reality than multiple equilibria. This is because PEA separates belief and the realization with time. It can account for both self-fulfilling and self defeating beliefs. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

85 Discussion PEA vs. Multiple Equilibria PEA is a more subtle notion of the relation between beliefs and reality than multiple equilibria. This is because PEA separates belief and the realization with time. It can account for both self-fulfilling and self defeating beliefs. It can detect weaker tendencies. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

86 Discussion PEA vs. Multiple Equilibria PEA is a more subtle notion of the relation between beliefs and reality than multiple equilibria. This is because PEA separates belief and the realization with time. It can account for both self-fulfilling and self defeating beliefs. It can detect weaker tendencies. Multiple equilibria is global and applies to all variables. PEA is measured with respect to some variable and some parameter mis-perception. Assumptions Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

87 Discussion Take Home Risk aversion heterogeneity banking. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

88 Discussion Take Home Risk aversion heterogeneity banking. Banker and speculator wealth and leverage dynamics generate counter-cyclical Sharpe Ratios. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

89 Discussion Take Home Risk aversion heterogeneity banking. Banker and speculator wealth and leverage dynamics generate counter-cyclical Sharpe Ratios. To account for high LTV news of disaster must arrive piecemeal. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

90 Discussion Take Home Risk aversion heterogeneity banking. Banker and speculator wealth and leverage dynamics generate counter-cyclical Sharpe Ratios. To account for high LTV news of disaster must arrive piecemeal. High risk premium is due to fear of financial system dysfunction. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

91 Discussion Take Home Risk aversion heterogeneity banking. Banker and speculator wealth and leverage dynamics generate counter-cyclical Sharpe Ratios. To account for high LTV news of disaster must arrive piecemeal. High risk premium is due to fear of financial system dysfunction. Self-defeating prophecies cause a 3-fold Prediction Error Amplification. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

92 Discussion Take Home Risk aversion heterogeneity banking. Banker and speculator wealth and leverage dynamics generate counter-cyclical Sharpe Ratios. To account for high LTV news of disaster must arrive piecemeal. High risk premium is due to fear of financial system dysfunction. Self-defeating prophecies cause a 3-fold Prediction Error Amplification. Only coordinated mistakes are amplified. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

93 Discussion Take Home Risk aversion heterogeneity banking. Banker and speculator wealth and leverage dynamics generate counter-cyclical Sharpe Ratios. To account for high LTV news of disaster must arrive piecemeal. High risk premium is due to fear of financial system dysfunction. Self-defeating prophecies cause a 3-fold Prediction Error Amplification. Only coordinated mistakes are amplified. Default on mortgages that were perceived as safe. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 42

94 Additional Material Table of Contents 7 Additional Material Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

95 Additional Material References I Pedro Bordalo, Nicola Gennaioli, and Andrei Shleifer. Diagnostic expectations and credit cycles. Technical report, National Bureau of Economic Research, Ana Fostel and John Geanakoplos. Leverage cycles and the anxious economy. The American Economic Review, 98(4): , Ana Fostel and John Geanakoplos. Why does bad news increase volatility and decrease leverage? Journal of Economic Theory, 147(2): , Mark Gertler and Nobuhiro Kiyotaki. Banking, liquidity, and bank runs in an infinite horizon economy. The American Economic Review, 105(7): , Robin Greenwood, Samuel G Hanson, and Lawrence J Jin. A model of credit market sentiment. Unpublished working paper, Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

96 Additional Material References II Hanno Lustig and Adrien Verdelhan. Business cycle variation in the risk-return trade-off. Journal of Monetary Economics, 59:S35 S49, Emi Nakamura, Jón Steinsson, Robert Barro, and José Ursúa. Crises and recoveries in an empirical model of consumption disasters. American Economic Journal: Macroeconomics, 5(3):35 74, Back Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

97 Additional Material Collateral Constraints - proof Assume that the only financial contracts allowed are of the form (promise, collateral)... Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

98 Additional Material Collateral Constraints - proof Assume that the only financial contracts allowed are of the form (promise, collateral)... And that there is no cost to default other than handing over the collateral. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

99 Additional Material Collateral Constraints - proof Assume that the only financial contracts allowed are of the form (promise, collateral)... And that there is no cost to default other than handing over the collateral. A promise to deliver the value of the collateral at down is safe for the lender and creates an Arrow up for the borrower. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

100 Additional Material Collateral Constraints - proof Assume that the only financial contracts allowed are of the form (promise, collateral)... And that there is no cost to default other than handing over the collateral. A promise to deliver the value of the collateral at down is safe for the lender and creates an Arrow up for the borrower. All possible portfolios can be mimicked with just safe loans and Arrow ups. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

101 Additional Material Collateral Constraints - proof Assume that the only financial contracts allowed are of the form (promise, collateral)... And that there is no cost to default other than handing over the collateral. A promise to deliver the value of the collateral at down is safe for the lender and creates an Arrow up for the borrower. All possible portfolios can be mimicked with just safe loans and Arrow ups. We can therefore focus on safe loans for convenience. Back Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

102 Additional Material Goods Markets Market clearing in the goods market states that c i (s) = w(s) s S. i {r,b,d} Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

103 Additional Material Goods Markets Market clearing in the goods market states that c i (s) = w(s) s S. i {r,b,d} The financial market clearing is i {r,b,d} i {r,b,d} B i (s) = 0 s S, (1 + r(s 1 ))B i (s 1 ) = 0 s S. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

104 Additional Material Housing Markets The stock of housing is fixed and the initial stock is set to one i {r,b,d} H i(0 1 ) = 1. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

105 Additional Material Housing Markets The stock of housing is fixed and the initial stock is set to one i {r,b,d} H i(0 1 ) = 1. Market clearing in housing rental market reads H i (s 1 ) = h i (s) s S. i {r,b,d} i {r,b,d} Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

106 Additional Material Housing Markets The stock of housing is fixed and the initial stock is set to one i {r,b,d} H i(0 1 ) = 1. Market clearing in housing rental market reads H i (s 1 ) = h i (s) s S. i {r,b,d} i {r,b,d} Market clearing in housing ownership market reads H i (s 1 ) = H i (s) s S. i {r,b,d} i {r,b,d} Back Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

107 Additional Material Equilibrium Definition Given initial endowments at node s, an equilibrium of node s is a mapping from all s that are decedent of s to values for p h (s ), R(s ), r(s ), ( ci (s ), h i (s ), H i (s ), B i (s ) ) i {r,b,d} such that s, i {r, b, d}: Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

108 Additional Material Equilibrium Definition Given initial endowments at node s, an equilibrium of node s is a mapping from all s that are decedent of s to values for p h (s ), R(s ), r(s ), ( ci (s ), h i (s ), H i (s ), B i (s ) ) i {r,b,d} such that s, i {r, b, d}: (c i (s ), h i (s ), H i (s ), B i (s )) are optimal for agent i. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

109 Additional Material Equilibrium Definition Given initial endowments at node s, an equilibrium of node s is a mapping from all s that are decedent of s to values for p h (s ), R(s ), r(s ), ( ci (s ), h i (s ), H i (s ), B i (s ) ) i {r,b,d} such that s, i {r, b, d}: (c i (s ), h i (s ), H i (s ), B i (s )) are optimal for agent i. Market clearing holds for the goods market, financial market, housing rental market, and housing ownership market. Back Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

110 Additional Material Financing Conditions Define Loan to Value (LTV) as LTV (s) p h(sd) + R(sd) (1 + r(s))p h (s). Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

111 Additional Material Financing Conditions Define Loan to Value (LTV) as LTV (s) p h(sd) + R(sd) (1 + r(s))p h (s). i {r, b, d} define Excess Collateral (EC) as the slack in the collateral constraint EC i (s) LTV (s)p h (s)h i (s) B i (s). Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

112 Additional Material Financing Conditions Define Loan to Value (LTV) as LTV (s) p h(sd) + R(sd) (1 + r(s))p h (s). i {r, b, d} define Excess Collateral (EC) as the slack in the collateral constraint EC i (s) LTV (s)p h (s)h i (s) B i (s). Define on equilibrium Cash On Hand (COH) as COH i (s) (p h (s) + R(s) (1 + r(s 1 ))LTV (s 1 )p h (s 1 ))H i (s 1 ) + W i w(s) + (1 + r(s 1 ))EC i (s 1 ). Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

113 Additional Material Leverage Leverage is defined as LEV i (s) = p h (s)h i (s) p h (s)h i (s)(1 LTV (s)) + EC i (s) i {r, b, d}. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

114 Additional Material Leverage Leverage is defined as LEV i (s) = p h (s)h i (s) p h (s)h i (s)(1 LTV (s)) + EC i (s) i {r, b, d}. For banks observable leverage includes also the safe loans taken by speculators as assets. It is defined as LEV b+r (s) = p h(s)h b (s) + LTV (s)p h (s)h r (s) EC r (s). p h (s)h b (s)(1 LTV (s)) + EC b (s) Back Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

115 Additional Material Asset Pricing The Risk Premium is defined as RP(s) = E s s +1 p h (s ) + R(s ) p h (s) r(s). Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

116 Additional Material Asset Pricing The Risk Premium is defined as RP(s) = E s s +1 p h (s ) + R(s ) p h (s) r(s). And finally, the Sharpe Ratio is defined as SR(s) = RP(s) ( [ ] ) 1. E ph (s )+R(s 2 2 ) s s +1 p h (s) r(s) RP(s) Back Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

117 Additional Material Off-Rational Expectations Payments The fact that off-rational expectations payments are bound by collateral requires us to restate the COH equations. COH r (s) W r w(s) + max(0,(p h (s) + R(s) (1 + r(s 1))LTV (s 1)p h (s 1))H r (s 1) COH b (s) + W b w(s) + (1 + r(s 1))EC r (s 1)) + max(0, (p h (s) + R(s) (1 + r(s 1))LTV (s 1)p h (s 1))H b (s 1) + (1 + r(s 1))EC b (s 1) + min(0,p h (s) + R(s) (1 + r(s 1))LTV (s 1)p h (s 1))H r (s 1) + (1 + r(s 1))EC r (s 1))) COH d (s) W d w(s) + (p h (s) + R(s) (1 + r(s 1))LTV (s 1)p h (s 1)H d (s 1) + (1 + r(s 1))EC d (s 1) + min(0, (p h (s) + R(s) (1 + r(s 1))LTV (s 1)p h (s 1))H b (s 1) + (1 + r(s 1))EC b (s 1) + min(0,p h (s) + R(s) (1 + r(s 1))LTV (s 1)p h (s 1))H r (s 1) + (1 + r(s 1))EC r (s 1))) Back Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

118 Additional Material Key Assumptions - Rational Expectations Section Most non-tradable wealth is held by very highly risk averse agents. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

119 Additional Material Key Assumptions - Rational Expectations Section Most non-tradable wealth is held by very highly risk averse agents. Risk aversion is heterogeneous. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

120 Additional Material Key Assumptions - Rational Expectations Section Most non-tradable wealth is held by very highly risk averse agents. Risk aversion is heterogeneous. Three discrete types can capture a continuum of risk aversion. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

121 Additional Material Key Assumptions - Rational Expectations Section Most non-tradable wealth is held by very highly risk averse agents. Risk aversion is heterogeneous. Three discrete types can capture a continuum of risk aversion. Type transitions are slow. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

122 Additional Material Key Assumptions - Rational Expectations Section Most non-tradable wealth is held by very highly risk averse agents. Risk aversion is heterogeneous. Three discrete types can capture a continuum of risk aversion. Type transitions are slow. State dependent contracts are hard to write. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13

123 Additional Material Key Assumptions - Rational Expectations Section Most non-tradable wealth is held by very highly risk averse agents. Risk aversion is heterogeneous. Three discrete types can capture a continuum of risk aversion. Type transitions are slow. State dependent contracts are hard to write. There is zero cost to seizing collateral. Yaniv Ben-Ami (Yale University) Banking Crises as Self-Defeating Prophecies February 22, / 13