## wpiea2020054-print-pdf — Section summaries and empirical implementation

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---

### Model overview and timing
- Two-period model starting at t = 0 with two possible macro-states at time t+1 = 1: Up (U) and Down (D).
- Agents: N banks, N+1 households, and the central bank.
- Banks heterogeneous in initial equity, risk-aversion γ_b, and default-cost coefficient λ_b.
- Households H1,...,HN borrow from banks B1,...,BN; household HN+1 supplies deposits to all banks.
- Timing:
  - Beginning of period 0: interbank, loan and deposit markets open; banks choose lending to households and interbank positions; central bank conducts OMOs to fix the interbank reference rate; then nature determines the macro-state.
  - Beginning of period 1: financial contracts settled; households and banks choose repayment fraction/probability of default; end of period markets re-open.

### Channels of systemic risk
- Interbank contagion:
  - Direct contagion via unsecured interbank market.
  - Banks choose to lend or borrow before the macro-state is realized; repayment fraction chosen after macro-state realization, transmitting default risk through interbank exposures.
  - Same default rate applied to each lender in the interbank market (for simplicity).
- Common exposure and fire sales:
  - Banks hold assets with correlated returns because loan losses depend on the macroeconomy.
  - Distressed banks curtail exposure → lower credit → depressed GDP → lower credit quality across the system.
- Minsky cycle (optimism channel):
  - Banks may perceive a higher subjective probability p of the good macro-state than the true probability.
  - Greater optimism increases allocation to risky assets and profit/loss volatility; common optimism amplifies systemic risk.
  - Proposition 1: ∂L_b_t/∂p > 0; ∂Π_b_t+1,U/∂p > 0; ∂Π_b_t+1,D/∂p < 0, implying larger expected shortfalls in profit when banks become more optimistic.

### Bank balance sheet, budget and key variables
- Stylized balance sheet: Assets = Loans L, Interbank loans LI, Market book A, Others ; Liabilities = Deposits D, Interbank deposits DI, Equity e.
- Notation highlights:
  - t∈T≡{0,1,2,...}, s∈S≡{U,D}, b∈B.
  - Exogenous: p_s, γ_b, λ_b, A_b_t, DI,b_t, LI,b_t, e_b_t, _b_t, r_b_t, rD,b_t, rA_t, ρ_t, δ_b_t+1,s, g_b,1, g_b,2, g_b,3, u_s,1, u_s,2.
  - Endogenous: U_b_t+1, Π_b_t+1,s, ∆Π_b_t+1,D, Λ_b_t+1,s, L_b_t, D_b_t, v_b_t+1,s, v_b,h_t+1,s, Y_t+1,s, M_t, ̃R_t+1.
- Budget constraint at time t (equation (2.4)):
  - L_b_t + LI,b_t + A_b_t = D_b_t + DI,b_t + e_b_t + _b_t
- Profit/loss at t+1 in state s (equation (2.5)):
  - Π_b_t+1,s = v_b,h_t+1,s L_b_t (1 + r_b_t) + ̃R_t+1,s LI,b_t (1 + ρ_t) + A_b_t (1 + rA_t) − v_b_t+1,s (D_b_t (1 + rD,b_t) + DI,b_t (1 + ρ_t)) − e_b_t − _b_t
- Expected shortfall in profit (equation (2.6)):
  - ∆Π_b_t+1,D = Π_b_t+1,U − Π_b_t+1,D
- Constraint on repayment fraction (as in source):
  - 0 ≤ v_b_t+1,s ≤ 1
- Unpaid obligations for deposits and interbank deposits (equation (2.8)):
  - Λ_b_t+1,s = (1 − v_b_t+1) (D_b_t (1 + rD,b_t) + DI,b_t (1 + ρ_t))
- Equity evolution (equation (2.9)):
  - e_b_t+1,s = e_b_t + Π_b_t+1,s − δ_b_t+1,s

### Banks’ optimization and equilibrium
- Banks maximize expected utility (equations (2.1)–(2.3)):
  - max_{σ_b_t={L_b_t, D_b_t, v_b_t+1,s}} E_t(U_b_t+1) = Σ_{s∈S} p_s (U_b(Π_b_t+1,s) − V_b(Λ_t+1,s))  (2.1)
  - U_b(Π) = Π − γ_b Π^2  (2.2)
  - V_b(Λ) = λ_b Λ^2  (2.3)
- Choice variables: leverage (deposits and interbank borrowing), credit to consumer loans, and repayment fraction v_b_t+1 (chosen after macro-state realization).
- Deposits and interbank loans have equal seniority; default rates identical for both.
- First-order condition wrt repayment rate for a lending bank (equation (2.14)):
  - 1 − 2 γ_b Π_b_t+1,s = 2 λ_b (1 − v_b_t+1,s) D_b_t (1 + rD,b_t), b∈B_L
  - Interpretation: LHS marginal benefit of default; RHS non-pecuniary marginal cost.
- Equilibrium (Definition 1): asset allocations and repayment rates such that:
  1. σ_b_t ∈ argmax_{σ_b_t ∈ B_b_t} E_t(U_b_t+1) for all b∈B
  2. Household repayments satisfy equation (2.10)
  3. GDP evolves by equation (2.11)
  4. Interbank market clears by equation (2.12)
  5. Banks’ expectations about interbank repayment rates satisfy equation (2.13)

### Households, GDP, and market-clearing
- Household repayment function (equation (2.10)):
  - ln(v_b,h_t+1) = g_b,1 + g_b,2 ln(Y_t+1,b) + g_b,3 Σ_{b∈B} ln(L_b_t)
  - Coefficients g_b,1, g_b,2, g_b,3 exogenous and calibrated.
- GDP function (equation (2.11)):
  - ln(Y_t+1,s) = u_s,1 + u_s,2 Σ_{b∈B} ln(L_b_t)
  - Parameters u_s,1 and u_s,2 calibrated.
- Interbank market clearing (equation (2.12)):
  - Σ_{b∈B} DI,b_t = M_t + Σ_{b∈B} LI,b_t
- Pooled interbank repayment rate (equation (2.13)):
  - ̃R_t+1,s = (Σ_{b∈B} v_b_t+1,s DI,b_t) / (Σ_{b∈B} DI,b_t)

### Financial distress, systemic risk measurement, and Minsky amplification
- Individual-level default endogenous: banks strategically choose v_b_t+1 balancing profit trade-offs and default penalty λ_b.
- Banking-sector interpretation:
  - Good macro-state: all banks healthy (no default).
  - Bad macro-state: at least one bank in distress; represents combination/average of remaining 2^N − 1 scenarios.
  - Due to contagion, all banks have positive default fractions in the bad state.
- Systemic risk measure: conditional expected shortfall ∆Π_b_t+1,D (equation (2.6)).
- Channel interaction:
  - Losses in a borrowing bank (v_b,h_t+1,D < 1) increase default fractions, affecting deposits and interbank lenders’ profits Π_b′_t+1,D and capital e_b′_t+1.
  - Common exposure + credit contraction → lower GDP → lower repayment rates system-wide (via equations (2.10) and (2.11)).
- Minsky optimism: higher subjective p leads banks to invest more in risky consumer loans, increasing profitability volatility and expected shortfalls.

### Empirical estimation: CIMDO method (Section 3 / 3.1 Method)
- Objective:
  - Infer unknown multivariate distribution p(l1,...,lM) of equity values from observed Probability of Distress (PoD) of each bank and a prior multivariate distribution q(l1,...,lM).
  - Cross-entropy approach recovers posterior distribution closest to prior but consistent with observed PoDs.
- Optimization formulation (bivariate example):
  - Minimize Kullback cross-entropy:
    - C[p(x,y),q(x,y)] = ∫∫ p(x,y) ln[p(x,y)/q(x,y)] dx dy  (3.1)
  - Constraints reproduce observed PoDs (Merton (1974) structural approach):
    - ∫∫ p(x,y) 1_{x>X_x^d} dx dy = PoD_x^t
    - ∫∫ p(x,y) 1_{y>X_y^d} dx dy = PoD_y^t  (3.2)
    - Posterior density must sum to 1.
- Closed-form posterior (Lagrange multipliers):
  - p̂(x,y) = q(x,y) exp{ −[1 + μ + (λ_x 1_{x>X_x^d}) + (λ_y 1_{y>X_y^d})] }  (3.3)
  - λ_x, λ_y, μ solve:
    - ∫∫ p̂(x,y) 1_{x>X_x^d} dx dy = PoD_x^t
    - ∫∫ p̂(x,y) 1_{y>X_y^d} dx dy = PoD_y^t
    - ∫∫ p̂(x,y) dx dy = 1  (3.4)
  - Numerical integration required to find multipliers.
- Intuition:
  - CIMDO adjusts prior with the exponential factor that depends on domain; tends to strengthen dependence when prior underestimates marginal PoDs.
- Prior calibration and inputs:
  - Prior calibrated using equity returns; stock market returns’ moments used to calibrate a prior’s t-distribution.
  - Observed PoDs crucial inputs; 1-year ahead CDS spreads used as proxies in this application.
  - Default thresholds X_x^d and X_y^d fixed to an average through time consistent with historical average PoD and prior distribution.
- Numerical implementation note:
  - Solving for λ_x, λ_y, μ requires numerical integration of posterior to match constraints.
  - CIMDO uses stock prices and EDFs to reconstruct joint distribution consistent with bank-specific PoDs and extract conditional expectations for systemic-risk exposure measures.

### Minsky cycle: model inversion and pre-crisis empirical inputs (Section 4.4)
- Model inverted to take empirical estimates of each bank’s conditional expected shortfall as input, solving for each bank’s subjective probability p_b1, b∈B.
- Empirical expected shortfalls in profits in June 2007 (CIMDO, change June 2006 → June 2007):
  - ΔΠ1 = 3.8507
  - ΔΠ2 = 1.3481
  - ΔΠ3 = 4.8913
  - ΔΠ4 = 3.0423
  - ΔΠ5 = 2.8821
- Model finds p_b1 > p_b0, ∀ b ∈ B (all five banks more optimistic in June 2006 than June 2005).
- Mechanism and implications:
  - Higher subjective probability of good state → higher risky loans L_b and higher leverage/investment in risky asset by individual banks.
  - Increased risk-taking raises frequency/fraction of default and creates externalities: banks do not internalize impacts on interbank repayment, future credit growth, or loan repayment rates.
  - Common shifts in beliefs strengthen common exposure and the feedback loop between credit growth and loan repayment rates.
- Fit and diagnostics:
  - Model predicted leverage ratios closer to actual data when Minsky cycle turned on.
- Baseline true good-state probability:
  - p = 0.94 (implies under binomial structure a crisis expected every twenty years).
  - Results robust for p > 0.91.

### Post-crisis period: simulations and calibrated results (Section 4.5)
- Simulations re-run using data after the financial crisis (2014–2016).
- CIMDO-predicted expected shortfalls (post-crisis calibration, Table 11):
  - ∆Π1 = 10.0377
  - ∆Π2 = 4.2823
  - ∆Π3 = 10.0982
  - ∆Π4 = 10.2032
  - ∆Π5 = 2.8422
- Model solution without Minsky effect, post-crisis (Table 12) — selected outputs:
  - ∆Π1 = 9.9097
  - ∆Π2 = 4.2798
  - ∆Π3 = 9.9848
  - ∆Π4 = 9.9161
  - ∆Π5 = 2.7696
- Model solution with Minsky effect, post-crisis (Table 13) — selected outputs:
  - ∆Π1 = 8.6901
  - ∆Π2 = 4.252
  - ∆Π3 = 8.477
  - ∆Π4 = 7.1881
  - ∆Π5 = 3.2704
- Banks’ implied subjective probabilities in post-crisis Minsky scenario (Table 13):
  - p1 = 0.9479
  - p2 = 0.9505
  - p3 = 0.9448
  - p4 = 0.9462
  - p5 = 0.9558
- Qualitative summary:
  - "According to the results in Table 13, 3 out of the 5 banks become less optimistic, while the other 2 become more optimistic."
  - "The systemic risk amplification due to the Minsky cycle of optimism is not as strong in the post-crisis period as in the pre-crisis period."
- Model-calibrated endogenous variables (selected post-crisis values, Table 11):
  - Π1U = 1.9951; Π1D = −8.0426; v1,h = 0.6476
  - Π2U = 1.0768; Π2D = −3.2055; v2,h = 0.7759
  - Π3U = 1.7022; Π3D = −8.396; v3,h = 0.778
  - Π4U = 1.4692; Π4D = −8.734; v4,h = 0.5937
  - Π5U = 0.6078; Π5D = −2.2345; v5,h = 0.7349
- Selected balance-sheet and macro observables used in calibration (Table 11):
  - M = 0.1762
  - ̃R = 0.899
  - ρ = 0.0454
  - YU = 182.248
  - YD = 164.0232
  - Manually selected probabilities and rates: p = 0.95; rA = 0.009; r1 = 0.0392; r1,d = 0.0098 (and analogous rates for other banks as listed)
- Interpretation:
  - Minsky cycle of optimism can account for a significant part of distress dependence structure pre-crisis; accounts for less loss in post-crisis period.
  - Post-crisis, Minsky-driven amplification of systemic risk weaker than pre-crisis.

*Source: wpiea2020054-print-pdf — Section 3, 4.4 and 4.5 summaries and empirical method (CIMDO).*

### Section 3 summarizes the empirical approach to computing the multivariate distribu-tion of

### Section 3 summarizes the empirical approach to computing the multivariate distribution of equity returns, and explains how it is implemented

### Model overview
- Two-period model starting at t= 0 with two possible macro-states at time t+1 = 1: the good state Up(U) and the bad state Down(D).
- Agents: N banks, N+ 1 households, and the central bank.
- Banks are heterogeneous (different initial equity, risk-aversion γb, and default-cost coefficient λb).
- Households H1,H2,...,HN borrow from banks B1,B2,...,BN; household HN+1 supplies deposits to all banks.
- Timing:
  - At beginning of period 0: interbank, loan and deposit markets open; banks choose lending to households and interbank positions; central bank conducts OMOs to fix the interbank reference rate; then nature determines the macro-state.
  - At beginning of period 1: financial contracts are settled; households and banks choose the probability/fraction of default (repayment rate); at end of period markets re-open.

### Channels of systemic risk
- Interbank contagion:
  - Direct contagion via unsecured interbank market.
  - Banks choose to lend or borrow before the macro-state is realized; likelihood/fraction of default is chosen after the macro-state is realized, transmitting default risk through interbank exposures.
  - For simplicity, the same default rate is applied to each lender in the interbank market.
- Common exposure and fire sales:
  - Banks hold assets with correlated returns because loan losses depend on the macroeconomy.
  - Distressed banks curtail exposure → lower credit → depressed GDP → lower credit quality across the system.
- Minsky cycle (optimism channel):
  - Banks may perceive a higher subjective probability p of the good macro-state than the true probability.
  - Greater optimism increases banks’ allocation to risky assets and increases profit/loss volatility; common optimism across banks amplifies systemic risk.
  - Proposition 1: ∂Lb_t/∂p >0; ∂Πb_t+1,U/∂p >0; ∂Πb_t+1,D/∂p <0, implying larger expected shortfalls in profit when banks become more optimistic.

### Bank balance sheet and key variables
- Stylized balance sheet (Table 1): Assets = Loans L, Interbank loans LI, Market book A, Others ; Liabilities = Deposits D, Interbank deposits DI, Equity e.
- Notation highlights (excerpt from Table 2):
  - Subscripts and superscripts: t∈T≡{0,1,2,...}, s∈S≡{U,D}, b∈B.
  - Exogenous variables include ps (probability of states), γb, λb, Abt, DI,b_t, LI,b_t, eb_t, b_t, rb_t, rD,b_t, rA_t, ρt, δb_t+1,s, gb,1, gb,2, gb,3, us,1, us,2.
  - Endogenous variables include Ub_t+1, Πb_t+1,s, ∆Πb_t+1,D, Λb_t+1,s, Lb_t, Db_t, vb_t+1,s, vb,h_t+1,s, Yt+1,s, Mt, ̃R_t+1.
- Budget constraint at time t (equation (2.4)):
  - Lb_t + LI,b_t + Ab_t = Db_t + DI,b_t + eb_t + b_t
- Profit/loss at t+1 in state s (equation (2.5)):
  - Πb_t+1,s = vb,h_t+1,s Lb_t (1 + rb_t) + ̃R_t+1,s LI,b_t (1 + ρt) + Ab_t (1 + rA_t) − vb_t+1,s (Db_t (1 + rD,b_t) + DI,b_t (1 + ρt)) − eb_t − b_t
- Expected shortfall in profit (equation (2.6)):
  - ∆Πb_t+1,D = Πb_t+1,U − Πb_t+1,D
- Constraint on repayment fraction (as in source):
  - 06v_b_t+1,s61
- Unpaid obligations for deposits and interbank deposits (equation (2.8)):
  - Λb_t+1,s = (1−v_b_t+1) (Db_t (1 + rD,b_t) + DI,b_t (1 + ρt))
- Equity evolution (equation (2.9)):
  - e_b_t+1,s = e_b_t + Π_b_t+1,s − δ_b_t+1,s

### Banks’ optimization and equilibrium
- Banks maximize expected utility (equations (2.1)–(2.3)):
  - max_{σ_b_t={Lb_t, Db_t, v_b_t+1,s}} E_t(U_b_t+1) = Σ_{s∈S} p_s (U_b(Π_b_t+1,s) − V_b(Λ_t+1,s))  (2.1)
  - U_b(Π) = Π − γ_b Π^2  (2.2)
  - V_b(Λ) = λ_b Λ^2  (2.3)
- Banks choose leverage (deposits and interbank borrowing), credit to consumer loans, and the probability/fraction of default v_b_t+1 (chosen after macro-state realization).
- Deposits and interbank loans have equal seniority; default rates are the same for both.
- First-order condition with respect to repayment rate for a lending bank (equation (2.14)):
  - 1 − 2 γ_b Π_b_t+1,s = 2 λ_b (1 − v_b_t+1,s) D_b_t (1 + rD,b_t),  b∈B_L
  - Interpretation: LHS marginal benefit of default; RHS non-pecuniary marginal cost.
- Equilibrium definition (Definition 1): set of asset allocations and repayment rates such that:
  1. σ_b_t ∈ argmax_{σ_b_t ∈ B_b_t} E_t(U_b_t+1) for all b∈B
  2. Household repayments satisfy equation (2.10)
  3. GDP evolves by equation (2.11)
  4. Interbank market clears by equation (2.12)
  5. Banks’ expectations about interbank repayment rates satisfy equation (2.13)

### Households, GDP, and market-clearing
- Household repayment function (equation (2.10)):
  - ln(v_b,h_t+1) = g_b,1 + g_b,2 ln(Y_t+1,b) + g_b,3 Σ_{b∈B} ln(L_b_t)
  - Coefficients g_b,1, g_b,2, g_b,3 are exogenous and calibrated.
- GDP function (equation (2.11)):
  - ln(Y_t+1,s) = u_s,1 + u_s,2 Σ_{b∈B} ln(L_b_t)
  - Parameters u_s,1 and u_s,2 to be calibrated.
- Interbank market clearing (equation (2.12)):
  - Σ_{b∈B} DI,b_t = M_t + Σ_{b∈B} LI,b_t
- Pooled interbank repayment rate (equation (2.13)):
  - ̃R_t+1,s = (Σ_{b∈B} v_b_t+1,s DI,b_t) / (Σ_{b∈B} DI,b_t)

### Financial distress, systemic risk measurement, and Minsky amplification
- Individual-level default is endogenous: banks choose fraction/probability of default (v_b_t+1) as a strategic decision driven by profit trade-offs and default penalty λ_b.
- Banking-sector interpretation:
  - Good macro-state: all banks healthy (no default).
  - Bad macro-state: at least one bank in distress; represents the combination/average of the remaining 2^N − 1 scenarios.
  - Due to contagion, all banks have positive default fractions in the bad state.
- Systemic risk measure used: conditional expected shortfall ∆Π_b_t+1,D (equation (2.6)) — difference in P&L between good and bad macro-states.
- Interaction of channels:
  - Losses in a borrowing bank (v_b,h_t+1,D < 1) increase default fractions, affecting deposits and interbank lenders’ profits Π_b′_t+1,D and capital e_b′_t+1.
  - Common exposure + credit contraction → lower GDP → lower repayment rates system-wide (via equations (2.10) and (2.11)).
- Minsky optimism: higher subjective p leads banks to invest more in risky consumer loans and increases profitability volatility and expected shortfalls (see Proposition 1).

### Empirical estimation of systemic risk (Section 3 summary)
- Goal: use empirical (cross-sectional) moments to inform the model channels of systemic risk.
- Cross-sectional moments: expected values for assets of one bank conditioning on asset value of another bank — estimated using a multivariate density capturing dependence among all banks’ asset values.
- Method: Consistent Information Multivariate Density Optimization (CIMDO) proposed by Segoviano (2006).
  - CIMDO constructs multivariate densities from a prior distribution and a reduced set of information (here, each bank’s probability of default).
  - Advantage: provides good estimates of dependence “at the tail”; non-parametric and general.
  - Ensures posterior distribution is the closest distribution to the prior that is consistent with information on probabilities of default.
  - CIMDO is based on the Kullback (1959) cross-entropy approach.
  - Prior work (Segoviano and Espinoza (2017)) shows CIMDO outperforms parametric distributions (t-distributions, mixture of normals) when focus on tail risk is essential.
- Empirical moments estimated via CIMDO are used to match the model’s implied moments (e.g., expected shortfalls) and recalibrate parameters for analysis of systemic risk channels and interactions.

*Source: wpiea2020054-print-pdf - Section 3 summarizes the empirical approach to computing the multivariate distribu-tion of equity returns, and explains how it is implemented.*

### 3.1    Method

### 3.1    Method

### CIMDO approach — objective
- CIMDO infers the unknown multivariate distribution of the equity value for a system M banks, p(l1,...,lM) (the posterior distribution), from the observed Probability of Distress (PoD) of each bank and from a prior multivariate distribution q(l1,...,lM).
- The cross-entropy approach recovers the distribution that is closest to the prior distribution but also consistent with the empirically observed probabilities of default of each bank.

### Optimization formulation
- Optimization problem (two-asset example, applicable when M >2):
  - Minimize over p(.,.) ∈ S the Kullback cross-entropy:
    - C[p(x,y),q(x,y)] = ∫∫ p(x,y) ln[p(x,y)/q(x,y)] dx dy (3.1)
  - q(x,y) ∈ R2 is the prior distribution and p(x,y) ∈ R2 is the posterior distribution.
  - C[p(x,y),q(x,y)] measures the distance between prior q and posterior p.

### Moment-consistency constraints (link to observed PoDs)
- Constraints impose that the posterior distribution reproduces the observed probability of distress for each bank (Merton (1974) structural approach):
  - ∫∫ p(x,y) 1_{x>X_x^d} dx dy = PoD_x^t
  - ∫∫ p(x,y) 1_{y>X_y^d} dx dy = PoD_y^t  (3.2)
  - where −x, −y are the equity annualized returns, 1 is the indicator function, and thresholds for banks x and y are X_x^d and X_y^d respectively.
- The posterior density must also sum to 1.

### Closed-form posterior (Lagrange multiplier solution)
- Segoviano and Espinoza (2017) give the posterior multivariate density:
  - p̂(x,y) = q(x,y) exp{ −[1 + μ + (λ_x 1_{x>X_x^d}) + (λ_y 1_{y>X_y^d})] } (3.3)
  - λ_x and λ_y are Lagrange multipliers corresponding to the PoD constraints; μ corresponds to the normalization constraint.
- The multipliers λ_x, λ_y, and μ solve the system (requires numerical integration):
  - ∫∫ p̂(x,y) 1_{x>X_x^d} dx dy = PoD_x^t
  - ∫∫ p̂(x,y) 1_{y>X_y^d} dx dy = PoD_y^t
  - ∫∫ p̂(x,y) dx dy = 1  (3.4)

### Intuition and dependence adjustment
- CIMDO adjusts the prior distribution with the factor exp{−[1 + μ + (λ_x 1_{x>X_x^d}) + (λ_y 1_{y>X_y^d})]}, which depends on the domain of the density.
- Segoviano and Espinoza (2017) show that CIMDO tends to strengthen dependence when marginal PoDs are underestimated by the prior.

### Prior calibration and inputs
- Prior density is calibrated using information on equity returns; stock market returns’ moments are used to calibrate a prior’s t-distribution.
- Observed PoDs are crucial inputs; for systemic risk applications in this paper, 1-year ahead CDS spreads are used as proxies.
- Default thresholds X_x^d and X_y^d are fixed to an average through time that is consistent with a historical average of the probability of default for each bank, and with the prior distribution.
  - (Footnote example: for bivariate standard normal prior, historical average distress thresholds set to X_x^d = Φ^{-1}(ᾱ_x) and X_y^d = Φ^{-1}(ᾱ_y), where Φ(·) is the standard normal cdf and ᾱ_x = 1−PoD_x and ᾱ_y = 1−PoD_y.)

### Numerical implementation note
- Solving for λ_x, λ_y, and μ requires numerical integration of the posterior density to match the three constraints in (3.4).
- CIMDO uses stock prices and EDFs as inputs to reconstruct the joint distribution consistent with bank-specific PoDs and extract conditional expectations used for systemic-risk exposure measures (see section 3.2).

*Source: wpiea2020054-print-pdf — 3.1 Method*

### 4.4    The Minsky cycle

### 4.4    The Minsky cycle

### Model inversion and calibrated inputs
- The model is inverted to take the empirical estimates of each bank’s conditional expected shortfall as input, and thus to solve for each bank’s subjective probability that a good state occurs (p_b1, b∈B).
- Figure 1 and Table 9 summarize the procedure. Table 10 in Appendix C presents the model results.
- The model finds that p_b1 > p_b0, ∀b ∈ B, i.e., according to the model, all the five banks in the sample had become more optimistic in June 2006 than they were in June 2005. This interpretation maps the empirically observed worsening of systemic risk (the increase in expected shortfalls) into rising subjective probabilities of the good state.

### Empirical expected shortfalls (CIMDO) used in the inversion
- Banks’ expected shortfalls in profits in June 2007, measured by CIMDO:
  - ΔΠ1 = 3.8507
  - ΔΠ2 = 1.3481
  - ΔΠ3 = 4.8913
  - ΔΠ4 = 3.0423
  - ΔΠ5 = 2.8821
- Notes: CIMDO generates the percentage change in equity value for each period. In this simulation, CIMDO predicts the change from June 2006 to June 2007. Given the balance sheet data of equity level for each bank in June 2006, the profit at the end of the period can be computed. The difference across states represents the expected shortfall in profit.

### Mechanism: optimism, risk-taking, and externalities
- When banks’ subjective probability of the good state increases, the model predicts:
  - Higher (risky) loans to households (L_b, b∈B) compared with when the Minsky cycle is turned off.
  - Rational increases in leverage and investment in the risky asset by individual banks if they believe the probability of a good state is high.
- Increased risk-taking raises the frequency/fraction of default, which acts as an externality:
  - Banks do not internalize the impact of their decisions on the repayment rate in the interbank market, on future credit growth, and on the repayment rate of loans.
  - When all banks simultaneously shift beliefs in the same direction, common exposure increases and the feedback loop between credit growth and loans’ repayment rates is stronger.

### Fit to observed leverage and model diagnostics
- The model’s prediction for the leverage ratios 19 is closer to actual data when the Minsky cycle is turned on.
  - Figure 9 shows the actual leverage ratios for the main UK banks along those predicted by the equilibrium model when the Minsky cycle is turned on and off.
  - This indicates that observed behavior in risk-taking can be represented relatively well by varying the subjective probability of the good macro-state occurring.

### Robustness to choice of the true good-state probability
- The baseline parameter choice is p = 0.94.
  - Under the binomial structure of uncertainty, this implies, on average, a crisis is expected every twenty years.
- The results remain when testing the model with different values for p > 0.91.
- Restatement in the source: The initial choice is p = 0.94, which implies under the binomial structure that on average every twenty years a crisis is expected. This is a quite optimistic view. We test the model with different values for p > 0.91, and all the results from above remain.
- The source notes that p is the probability of the good state in a binomial structure, thus 1 1−p represents the expected length of waiting period for a crisis to occur.

*Source: wpiea2020054-print-pdf — 4.4 The Minsky cycle*

### 4.5    Post-Crisis Period

### 4.5    Post-Crisis Period

### Summary of simulations and approach
- Simulations re-run using data after the financial crisis (2014-2016).
- Results are reported in Table 11, 12 and 13.
- CIMDO-predicted expected shortfalls in profits of each bank are lower than those predicted by the model that excludes the Minsky effect.
- To match CIMDO’s prediction, banks are allowed to update their beliefs (implied subjective probabilities) as in the previous section.

### Key empirical findings (post-crisis)
- CIMDO estimated expected shortfalls (∆Π) for the post-crisis calibration (Table 11):
  - ∆Π1 = 10.0377
  - ∆Π2 = 4.2823
  - ∆Π3 = 10.0982
  - ∆Π4 = 10.2032
  - ∆Π5 = 2.8422
- Model solution without Minsky effect, post-crisis (Table 12) — selected outputs:
  - ∆Π1 = 9.9097
  - ∆Π2 = 4.2798
  - ∆Π3 = 9.9848
  - ∆Π4 = 9.9161
  - ∆Π5 = 2.7696
- Model solution with Minsky effect, post-crisis (Table 13) — selected outputs:
  - ∆Π1 = 8.6901
  - ∆Π2 = 4.252
  - ∆Π3 = 8.477
  - ∆Π4 = 7.1881
  - ∆Π5 = 3.2704
- Banks’ implied subjective probabilities in the post-crisis Minsky scenario (Table 13):
  - p1 = 0.9479
  - p2 = 0.9505
  - p3 = 0.9448
  - p4 = 0.9462
  - p5 = 0.9558
- Qualitative result reported in the text: "According to the results in Table 13, 3 out of the 5 banks become less optimistic, while the other 2 become more optimistic."
- The text concludes: "The systemic risk amplification due to the Minsky cycle of optimism is not as strong in the post-crisis period as in the pre-crisis period."

### Model-calibrated endogenous variables (selected post-crisis values)
- Endogenously solved conditional P/L and default-related metrics in Table 11:
  - Π1U = 1.9951; Π1D = −8.0426; v1,h = 0.6476
  - Π2U = 1.0768; Π2D = −3.2055; v2,h = 0.7759
  - Π3U = 1.7022; Π3D = −8.396; v3,h = 0.778
  - Π4U = 1.4692; Π4D = −8.734; v4,h = 0.5937
  - Π5U = 0.6078; Π5D = −2.2345; v5,h = 0.7349
- Selected balance-sheet and macro observables used in calibration (Table 11):
  - M = 0.1762
  - ̃R = 0.899
  - ρ = 0.0454
  - YU = 182.248
  - YD = 164.0232
  - Manually selected probabilities and rates: p = 0.95; rA = 0.009; r1 = 0.0392; r1,d = 0.0098 (and analogous rates for other banks as listed)

### Interpretation and broader conclusions (as stated in the text)
- The paper highlights the importance of the Minsky cycle of optimism: when most banks become more optimistic at the same time, risk-taking increases, amplifying traditional sources of systemic risk.
- Empirical application: using UK banking-system data for 2005–2007 (pre-crisis) and 2014–2016 (post-crisis), the Minsky cycle of optimism can account for a significant part of the distress dependence structure pre-crisis, although it accounts for less loss in the more recent period.
- In the post-crisis period specifically, the Minsky-driven amplification of systemic risk is weaker than in the pre-crisis period.

*Source: wpiea2020054-print-pdf - 4.5    Post-Crisis Period*

### References

### wpiea2020054-print-pdf - References

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*Source: wpiea2020054-print-pdf - References*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020054-print-pdf.pdf_