## _wp1272 - introduction of new liquidity requirements, while several proposals have been recently advanced

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### Model and methodological contributions
- Main contribution: formulation of a dynamic banking model that jointly assesses capital regulation, liquidity requirements, and taxation in an environment where banks engage in maturity transformation and face multiple risks.
- Three novel model dimensions:
  - Dynamic transformation of short–term liabilities into longer–term partially illiquid assets with uncertain returns.
  - Explicit modeling of bank financial distress and three costly resolution options: asset liquidation (fire sale costs via asymmetric adjustment costs), issuance of fully collateralized bonds, or issuance of equity (with floatation costs).
  - Evaluation of regulatory and tax impacts on:
    - enterprise value V(x) = E(x) + F(x) − B as a metric of intermediation efficiency;
    - social value SV(x) = E(x) + D − B + G(x) + FC(x) as a welfare proxy in the risk-neutral setup.

### Benchmark environment and key calibration parameters
- Discount factor: β = 0.95.
- Risk–free rate: r_f = 2.5% (in one listing r_f annual risk–free rate on bonds: 2% — both figures are preserved as they appear in the source).
- Deposit rate: r_d = 0.
- Corporate tax marginal rates (base): τ_+ = 15%, τ_- = 5%.
- Bankruptcy/restructuring cost: γ = 0.10.
- Loan reimbursement fraction: δ = 20% (average maturity of outstanding loans = 1/δ − 1 = 4 years).
- Equity floatation cost (proportional): λ = 10%.
- Loan revenue function: π(L) = L^α with α = 0.90.
- Adjustment cost parameters (asymmetric): m_+ = 0.03, m_- = 0.04.
- Regulatory benchmarks: capital ratio k (base) = 4%; liquidity coverage ratio ` (base) = 0.2 (also reported as ` = 20% in the numerical parameterization).
- Alternative parameter values used in experiments: k = 12%; ` = 50%; τ_+ = 20%, τ_- = 7.5%; tax on non–deposit liabilities τ_B = 0.005; equity floatation cost increased to λ = 0.2 or set to λ = 0 in robustness checks; fire sale cost doubled: m_- from 0.04 to 0.08; maturity parameter δ reduced from 20% to 10% (average loan maturity increases from 4 years to 9 years).

### Bank structure and constraints (model mechanics)
- Assets: liquid one–period bond B_t yielding r_f and risky longer–term loans L_t with revenue π(L_t) Z_t.
- Assumption on loans: a proportion δ ∈ (0,1/2) of loans becomes due at t + 1.
- Loan adjustment costs: m(I_t) = |I_t|^2 (χ_{I_t>0} · m_+ + χ_{I_t<0} · m_-), with m_- > m_+ > 0.
- Liabilities: one-period deposits D_t (insured), and fully collateralized short–term bonds when issued (B_t < 0).
- Collateral constraint for issued bonds when B_t < 0 is given exactly by Equation (7) in the source (worst-case realizations Z_d and D_d enter).
- Deposit insurance: deposits fully insured; deposit insurance agency absorbs loss when bank defaults on deposits.
- Corporate taxation: τ(y_t) = τ_+ max{y_t,0} + τ_- min{y_t,0} with 0 ≤ τ_- ≤ τ_+ < 1.
- Equity issuance: seasoned equity issued at proportional flotation cost λ > 0.
- Insolvency: limited liability; in default assets transferred to deposit insurance agency net of reorganization costs γ·D_t; restructured bank formed with seed public capital K_{t+1} = D_u − D_{t+1} and initial investment only in risk–free bond (L_{t+1} = 0, B_{t+1} = D_u).

### Key theoretical and numerical findings — overview
- Three main sets of results:
  1. Capital requirements can reduce default risk and, if sufficiently mild, increase bank lending, enterprise value, and social value relative to an unregulated bank under deposit insurance. When capital requirements become too stringent, lending, efficiency, and social value decline — producing an inverted-U-shaped relationship between lending/efficiency/welfare and the stringency of capital requirements.
  2. Introduction of liquidity requirements reduces bank lending, enterprise value, and social value significantly. Reductions increase monotonically with the stringency of the liquidity requirement. Adding liquidity requirements to capital regulation can eliminate the benefits of mild capital requirements.
  3. Taxation: an increase in corporate income taxes reduces lending, bank efficiency, and social value while increasing government receipts. A tax on non–deposit liabilities (short–term debt in the model) leads to a larger decline in lending, efficiency, and social value than an equivalent increase in corporate taxation, but yields lower increases in government receipts. Under the benchmark calibration corporate income taxation is preferable to a tax on non–deposit liabilities.

### Detailed policy impacts and mechanisms
- Capital requirements (k):
  - Mild capital requirements (example base: k = 4%):
    - Can induce higher lending than the unregulated bank because banks respond by increasing retained earnings and issuing equity (subject to λ) and by holding less debt relative to the loan increase.
    - Result: enterprise value and social value can be higher than under no regulation; default probability is lower.
  - Stringent capital requirements (example: k = 12%):
    - Lead to lower lending, lower indebtedness, lower enterprise value, and lower social value compared to both the unregulated case and the mild-regulation case.
    - Intuition: high k makes retaining earnings and/or issuing equity prohibitively costly relative to reducing lending; hence diminished maturity transformation and lower efficiency.
  - Implication: existence of an optimal bank–specific regulatory capital level; relationship between k and lending/efficiency/welfare is inverted U-shaped.

- Liquidity requirements (`):
  - Liquidity coverage ratio (`) modeled as: (δL + Z_d π(L) − τ(y_min) + B(1 + r_f)) / (D(1 + r_d) − D_d) ≥ ` (Equation (25) exact form in source).
  - Effect: liquidity requirements constrain maturity transformation directly, forcing higher holdings of liquid assets or bonds and thereby reducing loan investment and short–term debt issuance.
  - When added to capital requirements, liquidity constraints often dominate and can eliminate the positive effects of mild capital regulation.
  - Result: marked reductions in lending, enterprise value, and social value. Average book capital ratios become much higher due to high bond holdings and lower loan denominators, producing a safe but inefficient banking configuration.

- Taxation:
  - Increase in corporate income taxes (benchmark experiment: τ_+ from 15% to 20%, τ_- from 5% to 7.5%):
    - Reduces lending, enterprise value, and social value; increases government value via higher tax receipts.
  - Tax on non–deposit liabilities (τ_B = 0.005):
    - Causes a sharper decline in lending and indebtedness than an increase in corporate taxes.
    - Yields lower government receipts than the comparable corporate tax increase in benchmark calibration.
    - When liquidity requirements are binding, the impact of τ_B is muted.
  - Policy implication: corporate income taxation is preferable to a tax on non–deposit liabilities under the benchmark calibration because it raises more revenue with lower efficiency/welfare costs.

### Simulation, steady state evidence, and numerical implementation
- Simulation design:
  - Simulate 10,000 shock paths of 100 annual periods; drop first 50 periods; report mean and standard deviation over 10,000 × 50 observations.
- State and solution method:
  - Continuous–decision, infinite–horizon Markov Decision Process solved by value iteration using discretization of ξ(t) = (Z(t), log D(t)) and control grids for L and B.
  - Implementation details: L_u = 10, B_d = −5, B_u = 5; use 9 Z points, 9 D points, N_L = 27, N_B = 36; termination tolerance = 10^{−5}; Monte Carlo simulation Ω = 10,000 paths, T = 100 years (drop first 50).
- Numerical parameterization (Table I — base case):
  - κ_Z: 0.88
  - σ_Z: 0.0139
  - Z: 0.0717
  - κ_D: 0.95
  - σ_D: 0.0209
  - D: $2
  - ρ: −0.85
  - β: 0.95
  - r_f: 2% (annual risk–free rate on bonds)
  - r_d: 0%
  - τ_+: 15%
  - τ_-: 5%
  - δ: 20%
  - γ: 0.1
  - λ: 0.1
  - α: 0.90
  - m_+: 0.03
  - m_-: 0.04
  - k: 4%
  - `: 20%
  - τ_B: 0

- Key numerical comparative medians (Table IV highlights, base parameterization unless noted):
  - Unregulated base:
    - Loan (book) = 4.78
    - Net Bond Holdings (book) = −3.48
    - Bank Capital (book) = −0.70
    - Equity (mkt) = 4.49
    - Deposits (mkt) = 1.89
    - Enterprise Value (mkt) = 9.88
    - Government Value (mkt) = 0.54
    - Social value (mkt) = 10.56
    - Default/Closure Rate (pct) = 5.34
  - Capital (k = 12% example column):
    - Loan (book) = 6.37
    - Net Bond Holdings (book) = −3.89
    - Bank Capital (book) = 0.47
    - Equity (mkt) = 4.87
    - Enterprise Value (mkt) = 10.71
    - Government Value (mkt) = 0.88
    - Social value (mkt) = 11.72
    - Default/Closure Rate (pct) = 0.00
  - Capital & liquidity (base k = 4%, ` = 20% example column):
    - Loan (book) = 2.68
    - Net Bond Holdings (book) = 0.16
    - Bank Capital (book) = 0.82
    - Equity (mkt) = 3.69
    - Enterprise Value (mkt) = 5.43
    - Government Value (mkt) = 0.37
    - Social value (mkt) = 5.91
    - Default/Closure Rate (pct) = 0.00
  - Closure rule comparisons (Table VII medians):
    - Unregulated: Loan (book) = 4.78; Enterprise Value (mkt) = 9.88; Social value (mkt) = 10.56; Default/Closure Rate (pct) = 5.34.
    - k = 0 (nonnegative book equity): Loan (book) = 6.14; Enterprise Value (mkt) = 10.58; Social value (mkt) = 11.56; Default/Closure Rate (pct) = 21.75.
    - Closure rule: Loan (book) = 5.60; Enterprise Value (mkt) = 10.76; Social value (mkt) = 11.63; Default/Closure Rate (pct) = 21.75.

### Robustness and sensitivity (selected experiments)
- Equity floatation cost (λ):
  - λ = 0 strengthens the inverted-U relationship (easier equity issuance sustains lending under higher k).
  - Increasing λ from 0.1 to 0.2 reduces lending, enterprise, and social values; effect smaller relative to retained earnings management.
- Fire sale costs (m_-):
  - Doubling m_- from 0.04 to 0.08 increases lending, enterprise value, and social value as banks retain more earnings to avoid costly liquidations.
- Maturity mismatch (δ):
  - Reducing δ from 20% to 10% (longer loan maturity) leads to stronger maturity transformation in capital-only setting (higher loans and indebtedness).
  - When liquidity requirements are imposed, reductions in lending, enterprise value, and social value are more pronounced for greater maturity mismatch.

### Policy takeaways and recommendations
- Capital regulation:
  - Well–calibrated, mild capital requirements (example benchmark k = 4%) can reduce default risk and increase lending, enterprise value, and social value relative to an unregulated bank under deposit insurance.
  - Overly stringent capital requirements (example k = 12%) can reverse these benefits — indicating the existence of an optimal, bank-specific capital level.
- Liquidity requirements:
  - Liquidity requirements substantially curb banks’ maturity transformation and produce large reductions in lending, enterprise value, and social value.
  - If liquidity requirements are justified to correct externalities from short–term debt reliance, their benefits must at least offset the substantial costs documented.
- Taxation:
  - Corporate income taxation is less damaging to lending and welfare and raises more government revenue—under benchmark calibration—than a tax on non–deposit liabilities.
  - Taxes on non–deposit liabilities tend to reduce lending more sharply and generate lower revenue increases in the model.
- Interactions and heterogeneity:
  - Joint assessment is crucial: liquidity requirements can nullify efficiency/welfare gains from modest capital requirements; joint welfare implications differ from isolated policy assessments.
  - Design of regulation should account for bank heterogeneity (maturity structure, fire-sale costs, equity issuance frictions) because optimal capital and liquidity prescriptions are likely bank-specific.

*IMF working paper content unit _wp1272 (introduction of new liquidity requirements, while several proposals have been recently advanced) — extracted content provided in the prompt.*

### introduction of new liquidity requirements, while several proposals have been recently advanced

### _wp1272 - introduction of new liquidity requirements, while several proposals have been recently advanced

### Model and methodological contributions
- Main contribution: formulation of a dynamic banking model that jointly assesses capital regulation, liquidity requirements, and taxation in an environment where banks engage in maturity transformation and face multiple risks.
- Three novel model dimensions:
  - Dynamic transformation of short–term liabilities into longer–term partially illiquid assets with uncertain returns.
  - Explicit modeling of bank financial distress and three costly resolution options: asset liquidation (fire sale costs via asymmetric adjustment costs), issuance of fully collateralized bonds, or issuance of equity (with floatation costs).
  - Evaluation of regulatory and tax impacts on (i) enterprise value (V(x) = E(x) + F(x) − B) as a metric of intermediation efficiency and (ii) social value (SV(x) = E(x) + D − B + G(x) + FC(x)) as a welfare proxy in the risk-neutral setup.

### Benchmark environment and key calibration parameters
- Discount factor: β = 0.95.
- Risk–free rate: r_f = 2.5%.
- Deposit rate: r_d = 0.
- Corporate tax marginal rates (base): τ_+ = 15%, τ_- = 5%.
- Bankruptcy/restructuring cost: γ = 0.10.
- Loan reimbursement fraction: δ = 20% (average maturity of outstanding loans = 1/δ − 1 = 4 years).
- Equity floatation cost (proportional): λ = 10%.
- Loan revenue function: π(L) = L^α with α = 0.90.
- Adjustment cost parameters (asymmetric): m_+ = 0.03, m_- = 0.04.
- Regulatory benchmarks: capital ratio k (base) = 4%; liquidity coverage ratio ` (base) = 0.2.
- Alternative parameter values used in experiments: k = 12%; ` = 50%; τ_+ = 20%, τ_- = 7.5%; tax on non–deposit liabilities τ_B = 0.005; equity floatation cost increased to λ = 0.2 or set to λ = 0 in robustness checks; fire sale cost doubled: m_- from 0.04 to 0.08; maturity parameter δ reduced from 20% to 10% (average loan maturity increases from 4 years to 9 years).

### Bank structure and constraints (model mechanics)
- Assets: liquid one–period bond B_t yielding r_f and risky longer–term loans L_t with revenue π(L_t) Z_t.
- Assumption on loans: a proportion δ ∈ (0,1/2) of loans becomes due at t + 1.
- Loan adjustment costs: m(I_t) = |I_t|^2 (χ_{I_t>0} · m_+ + χ_{I_t<0} · m_-), with m_- > m_+ > 0 (capturing costly reversibility/fire-sale costs).
- Liabilities: one-period deposits D_t (insured), and fully collateralized short–term bonds when issued (B_t < 0).
- Collateral constraint for issued bonds when B_t < 0 is given exactly by Equation (7) in the source (worst-case realizations Z_d and D_d enter).
- Deposit insurance: deposits fully insured; deposit insurance agency absorbs loss when bank defaults on deposits.
- Corporate taxation: τ(y_t) = τ_+ max{y_t,0} + τ_- min{y_t,0} with 0 ≤ τ_- ≤ τ_+ < 1.
- Equity issuance: seasoned equity issued at proportional flotation cost λ > 0.
- Insolvency: limited liability; in default assets transferred to deposit insurance agency net of reorganization costs γ·D_t; restructured bank formed with seed public capital K_{t+1} = D_u − D_{t+1} and initial investment only in risk–free bond (L_{t+1} = 0, B_{t+1} = D_u).

### Key theoretical and numerical findings — overview
- Three main sets of results are obtained from model solution and simulations:
  1. Capital requirements can reduce default risk and, if sufficiently mild, increase bank lending, enterprise value, and social value relative to an unregulated bank under deposit insurance. However, when capital requirements become too stringent, lending, efficiency, and social value decline — producing an inverted-U-shaped relationship between lending/efficiency/welfare and the stringency of capital requirements.
  2. Introduction of liquidity requirements reduces bank lending, efficiency (enterprise value), and social value significantly. Reductions increase monotonically with the stringency of the liquidity requirement. Adding liquidity requirements to capital regulation can eliminate the benefits of mild capital requirements.
  3. Taxation: an increase in corporate income taxes reduces lending, bank efficiency, and social value while increasing government receipts. A tax on non–deposit liabilities (short–term debt in the model) leads to a larger decline in lending, efficiency, and social value than an equivalent increase in corporate taxation, but yields lower increases in government receipts. Under the benchmark calibration corporate income taxation is preferable to a tax on non–deposit liabilities.

### Detailed policy impacts and mechanisms
- Capital requirements (k):
  - Mild capital requirements (example base: k = 4%):
    - Can induce higher lending than the unregulated bank because banks respond by increasing retained earnings and issuing equity (subject to λ) and by holding less debt relative to the loan increase.
    - Result: enterprise value and social value can be higher than under no regulation, especially when fire sale costs (m_-) are larger.
    - Default probability is lower than the unregulated bank.
  - Stringent capital requirements (example: k = 12%):
    - Lead to lower lending, lower indebtedness, lower enterprise value, and lower social value compared to both the unregulated case and the mild-regulation case.
    - Intuition: high k makes retaining earnings and/or issuing equity prohibitively costly relative to reducing lending; hence diminished maturity transformation and lower efficiency.
  - Implication: existence of an optimal bank–specific regulatory capital level; relationship between k and lending/efficiency/welfare is inverted U-shaped.

- Liquidity requirements (`):
  - Liquidity coverage ratio (`) modeled as: (δL + Z_d π(L) − τ(y_min) + B(1 + r_f)) / (D(1 + r_d) − D_d) ≥ ` (Equation (25) exact form in source).
  - Effect: liquidity requirements constrain maturity transformation directly (force higher holdings of liquid assets or bonds), thereby reducing loan investment and short–term debt issuance.
  - When added to capital requirements, liquidity constraints often dominate and can eliminate the positive effects of mild capital regulation.
  - Result: marked reductions in lending, enterprise value, and social value. Average book capital ratios become much higher (over–inflated) due to high bond holdings and lower loan denominators, producing a safe but inefficient banking configuration.

- Taxation:
  - Increase in corporate income taxes (benchmark experiment: τ_+ from 15% to 20%, τ_- from 5% to 7.5%):
    - Reduces lending, enterprise value, and social value (standard negative income effects).
    - Increases government value via higher tax receipts.
  - Tax on non–deposit liabilities (τ_B = 0.005):
    - Causes a sharper decline in lending and indebtedness than an increase in corporate taxes.
    - Yields lower government receipts than the comparable corporate tax increase in benchmark calibration.
    - When liquidity requirements are binding (banks hold no or positive net bond holdings on average), the impact of τ_B is muted.
  - Policy implication: corporate income taxation is preferable to a tax on non–deposit liabilities under the benchmark calibration because it raises more revenue with lower efficiency/welfare costs.

### Simulation and steady state evidence
- Simulation design:
  - Simulate 10,000 shock paths of 100 annual periods; drop first 50 periods; report mean and standard deviation of policies and metrics over the panel of 10,000 × 50 observations.
- Key steady state comparative outcomes (base calibration unless noted):
  - Unregulated bank:
    - Optimal policy is pro-cyclical lending (loans increase with favorable credit Z and higher D′).
    - Average capital and liquidity coverage ratios can be negative in worst-case definitions, implying reliance on forced liquidation in bad states.
  - Bank with mild capital requirement (k = 4%):
    - Invests more in loans and holds more debt than unregulated bank, but debt increases less than loans.
    - Maintains capital ratios above the regulatory minimum on average (precautionary retained earnings).
    - Enterprise value and social value are higher than the unregulated case; default probability lower.
  - Bank with capital requirement plus liquidity requirement (` = 20% base):
    - Loans and indebtedness shrink substantially relative to capital-only case.
    - Enterprise, government, and social values all decline significantly relative to capital-only case.
    - Increasing k from 4% to 12% when ` = 20% produces negligible additional change because liquidity requirement already binds.
    - Increasing ` further (e.g., to 50%) reduces loans and values further.

### Robustness and sensitivity (selected experiments)
- Equity floatation cost (λ):
  - Eliminating equity issuance costs (λ = 0) strengthens the inverted-U relationship (banks can more easily raise equity under higher k and sustain lending).
  - Increasing λ from 0.1 to 0.2 reduces lending, enterprise, and social values, but the effect is relatively small compared to retained earnings management effects.
- Fire sale costs (m_-):
  - Doubling m_- from 0.04 to 0.08 increases lending, enterprise value, and social value (banks optimally retain more earnings and invest more to avoid costly future liquidations). Results hold for capital-only and capital+liquidity cases.
- Maturity mismatch (δ):
  - Reducing δ from 20% to 10% (longer loan maturity, higher maturity mismatch) leads to stronger maturity transformation in capital-only setting (higher loans and indebtedness).
  - When liquidity requirements are imposed, the reductions in lending, enterprise value, and social value are more pronounced for banks with greater maturity mismatch.

### Policy takeaways and recommendations
- Capital regulation:
  - Well–calibrated, mild capital requirements (example benchmark k = 4%) can simultaneously reduce default risk and increase lending, enterprise value, and social value relative to an unregulated bank under deposit insurance.
  - Overly stringent capital requirements (example k = 12%) can reverse these benefits — producing lower lending and welfare — indicating the existence of an optimal, bank-specific capital level.
- Liquidity requirements:
  - Liquidity requirements substantially curb the maturity transformation role of banks and produce large reductions in lending, enterprise value, and social value.
  - If liquidity requirements are justified to correct externalities from short–term debt reliance, the benefits must at least offset the substantial costs documented.
- Taxation:
  - Corporate income taxation is less damaging to lending and welfare and raises more government revenue—under benchmark calibration—than a tax on non–deposit liabilities.
  - Taxes on non–deposit liabilities tend to reduce lending more sharply and generate lower revenue increases in the model.
- Interactions:
  - The joint assessment is crucial: liquidity requirements can nullify the efficiency/welfare gains from modest capital requirements; the joint welfare implications differ from isolated policy assessments.
  - Design of regulation should account for bank heterogeneity (maturity structure, fire-sale costs, equity issuance frictions) because optimal capital and liquidity prescriptions are likely bank-specific.

*Italicized source attribution: IMF working paper content unit _wp1272 (introduction of new liquidity requirements, while several proposals have been recently advanced) — extracted content provided in the prompt.*

### References

### References and Appendix (Content Unit: _wp1272 - References)

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- Shackelford, D., D. Shaviro, and J. Slemrod, 2010, Taxation and the Financial Sector, NYU School of Law, Law and Economics Research Working paper.
- Stokey, N., and R. E. Lucas, 1989, Recursive Methods in Economic Dynamics. (Harvard University Press Cambridge, MA).
- Stulz, R. M., 1984, Optimal Hedging Policies, Journal of Financial and Quantitative Analysis19, 127–140.
- Tauchen, G., 1986, Finite State Markov-chain Approximations to Univariate and Vector Autoregressions, Economics Letters20, 177–181.
- Van den Heuvel, S. J., 2008, The Welfare Cost of Bank Capital Requirements, Journal of Monetary Economics55, 298–320.
- Van den Heuvel, S. J., 2009, The Bank Capital Channel of Monetary Policy, Working paper, Federal Reserve Board.
- Zhu, H., 2008, Capital Regulation and Banks’ Financial Decisions, International Journal of Central Banking4, 165–211.

### Appendix A. Properties of the unregulated bank program
- Compactness of the feasible set of the bank can be shown as follows.
- Given the strict concavity of π(L), there exists a level L_u such that π(L_u) Z_u − rL_u = 0, where r is the cost of capital of the marginal dollar raised either through deposits or short term financing.
- Any investment L > L_u would be unprofitable. This establishes an upper bound on the feasible set of L, given by [0, L_u] for some L_u.
- With an upper bound on L, and because the stochastic process D has compact support, the collateral constraint sets a lower bound B_d (i.e., an upper bound on bond issuance). Specifically, this is obtained by putting D_d in place of D_t and L_u in place of L_t in equation (7).
- An upper bound on B can be obtained assuming that the proceeds from risk–free investments made by the bank are taxed at a higher rate than the personal tax rate and that floatation costs are positive. Assume that the current deposits D are all invested in short term bonds, B, with no investments in loans.
- To increase investment in bonds by one dollar, the bank must raise equity capital. A shareholder thus incurs a cost 1 + λ, where λ is the floatation cost. This additional dollar is invested at the risk–free rate, so that at the end of the year, the proceeds of this investment that can be distributed are (1 + r_f (1−τ_+)).
- Alternatively, the shareholder can invest 1 + λ in a risk–free bond, obtaining (1 + λ)(1 + r_f). Because τ_+ ≥ 0 and λ ≥ 0, then (1 + λ)(1 + r_f) ≥ (1 + r_f (1−τ_+)), there is no incentive of the bank to have a cash balance that is larger than D as long as either λ or τ_+ are strictly positive.
- If shareholders are taxed on their investment proceeds at a rate τ_p, they obtain (1 + λ)(1 + r_f (1−τ_p)) from their investment in the risk–free asset. If τ_p ≤ τ_+, then (1 + λ)(1 + r_f (1−τ_p)) > (1 + r_f (1−τ_+)), and the bank has no incentive to increase the investment in risk–free bonds beyond D.
- Moreover, if floatation costs associated with equity issuance are strictly increasing in the amount issued, no assumption about differential tax rates are needed to establish an upper bound on B.

*Content derived from the PDF chapter/section titled "References" and "Appendix A. Properties of the unregulated bank program" from the supplied source.*

### conclusion, the feasible set of the bank can be assumed to be [0,L

### _wp1272 - conclusion, the feasible set of the bank can be assumed to be [0,L_u]×[B_d,B_u]

### Existence and properties of the bank value function
- A unique value function E(x) = E(L,B,D,Z,D′) satisfies equation (15).
- E is continuous, increasing, and differentiable in all its arguments, and concave in L.
- Existence and uniqueness follow from the Contraction Mapping Theorem (Theorem 3.2 in Stokey and Lucas (1989)).
- Continuity, monotonicity, and concavity in L follow from Theorem 3.2 and Lemma 9.5 in Stokey and Lucas (1989).
- Continuity and monotonicity in B, D, Z, and D′ follow from continuity and monotonicity of e in Z and D′ and the monotonicity of the Markov transition function of (Z,D).

### Bank closure rule (section B)
- Closure rule specification:
  - Bank is closed at time t if ex–post net asset value v_t = L_t + B_t − D_t + y_t − τ(y_t) = K_t + y_t − τ(y_t) < 0. (Equation (32))
  - Closure threshold is higher than insolvency threshold; the rule rules out bank insolvency.
  - Current equity value E(x) > 0 is expropriated from shareholders and transferred to the government.
  - New capital K_{t+1} = D_u − D_{t+1} is injected.
  - Government direct restructuring costs are assumed γ = 0 in illustrated case.
- Government net payoff recursion under closure rule:
  - G(x) = (1−Δ(x)) ( τ(y′) + βE[G(x′)] ) + Δ(x) (E(x) − γD − D_u + D_{t+1}). (Equation (33))
  - Δ denotes the default indicator at x. If Δ = 1, government incurs direct bankruptcy costs γD, expropriates E(x), and injects K_{t+1} = D_u − D_{t+1}.
- When B_t < 0 is fully collateralized and the bank is taken over:
  - Bond holders, (old) depositors, and the government are paid in full.
  - Financing shortfall (net obligations minus available liquid funds) is (1 + r_d)D_t − (1 + r_f)B_t + τ(y_t) − (D_{t+1} + π(L_t)Z_t).
  - Loan liquidation/portfolio reduction equation:
    - L_t − L_{t+1} − m(L_t(1−δ) − L_{t+1}) = (1 + r_d)D_t − (1 + r_f)B_t + τ(y_t) − (D_{t+1} + π(L_t)Z_t).
  - New loan level L^*_ {t+1} is determined and is positive because of collateral constraint.
  - Indirect social cost incurred due to loan fire sales costs m(L_t(1−δ) − L^*_{t+1}).
- Comparative outcomes:
  - Closure rule has an impact similar to a requirement to hold non–negative book equity (k = 0).
  - Relative to unregulated bank, closure rule increases loans, enterprise and social values under assumption of no bank reorganization costs.
  - Closure rule results in very high rate of government intervention in closing and reopening banks, suggesting it may be more costly than capital requirements when positive reorganization costs are more realistic.

### Numerical solution algorithm (section C)
- Model: continuous–decision, infinite–horizon Markov Decision Process solved by value iteration (successive approximations of Bellman equation (15)).
- State vector ξ(t) = (Z(t), log D(t)) has VAR form:
  - ξ(t) = c + K ξ(t−1) + ε(t). (Equation (34))
  - ε = (ε_Z, ε_D) is bivariate Normal with mean zero and covariance Σ.
  - c = ((1−κ_1) ξ_1, (1−κ_2) ξ_2)^T, K = diag(κ_1, κ_2), Σ = [[σ_1^2, σ_1 σ_2 ρ], [σ_1 σ_2 ρ, σ_2^2]].
- Discretization and transition matrix:
  - Continuous ξ(t) approximated by discrete-state ˆξ(t) on finite grid {X_1,...,X_n} ⊂ R^2.
  - Π_{i,j} = Prob{ˆξ(t+1) ∈ X_j | ˆξ(t) = X_i} = ∫_{X′_j} φ(ξ,0,Σ) dξ where X′_j = X_j − c − K ξ(t).
  - Integral computed by Monte–Carlo integration; grid points selected based on equal weighting from univariate normal CDF.
- Control discretization and feasible sets:
  - Feasible intervals: loans [0, L_u], bonds [B_d, B_u] (with B_d < 0 < B_u), set so they are never binding for equity maximizing program.
  - Loan grid: ˜L = { ˜L_j = L_u (1−δ)^j | j = 1,...,N_L − 1 } ∪ {L_{N_L} = 0}.
  - Bond grid: [B_d, B_u] discretized into N_b equally–spaced values forming ˜B.
  - Discrete state x = (ξ, L, B).
- Bellman problem solved numerically:
  - E(x) = max { 0, max_{(L′,B′) ∈ A(D)} { e(x,L′,B′) + β E[E(x′)] } }, where e(x,L,B) from equation (10) and A(D) is case-specific feasible set.
  - Iteration stops when ‖E_{j+1} − E_j‖ < tolerance.
- Implementation details and simulation:
  - For parameters in Table I: L_u = 10, B_d = −5, B_u = 5.
  - Use 9 points for Z, 9 points for D, N_L = 27, N_B = 36.
  - Termination tolerance = 10^{−5}.
  - Monte Carlo simulation of Ω = 10,000 paths, T = 100 years (steps); drop first 50 steps to limit initial-condition dependence.
  - Initial simulated ξ: Z(0) = Z and D(0) = D_d. Initial feasible choices L(0) = 0 and B(0) = D_u so initial bank capital is D_u − D_d.

### Numerical parameterization (Table I) — base case model parameters
- κ_Z annual persistence of the credit shock: 0.88
- σ_Z annual conditional std. dev. of the credit shock: 0.0139
- Z unconditional average of the credit shock: 0.0717
- κ_D annual persistence of the log of deposits: 0.95
- σ_D annual conditional std. dev. of the log of deposits: 0.0209
- D unconditional average of deposits: $2
- ρ correlation between log–deposit and credit shock: −0.85
- β annual discount factor: 0.95
- r_f annual risk–free rate on bonds: 2%
- r_d annual rate on deposits: 0%
- τ_+ corporate tax rate for positive earnings: 15%
- τ_− corporate tax rate for negative earnings: 5%
- δ annual percentage of reimbursed loan: 20%
- γ bankruptcy costs: 0.1
- λ flotation cost for equity: 0.1
- α return to scale for loan investment: 0.90
- m_+ unit price for loan investment: 0.03
- m_− unit price for loan fire sales: 0.04
- k percentage of loans for capital regulation: 4%
- ` liquidity coverage ratio: 20%
- τ_B tax rate on uninsured liabilities: 0

### Key numerical findings and comparative results (selected)
- Figures and tables characterize policy and value effects across three regimes: Non–Regulated; Capital Requirement; Capital & Liquidity Requirement.
- Policy responses (Figure 3):
  - Optimal B^* and L^*/L vary across liquidity shock (D′ − D) and credit shock Z; base steady state used: credit shock = 0.0717, D = 2, B = 0, L = 4.1 so K = 2.1.
- Value impacts (Figure 4):
  - Enterprise value and social value under regulations shown as proportions of unregulated case across shocks; values computed for base parameterization using 9 Z points, 9 D points, 27 L points, 36 B points (Table I).
- Selected quantitative summaries from tables (medians across Monte Carlo runs, base parameterization unless stated):
  - Table II: Average capital ratio ((L^* + B^* − D′)/L^*) reported as function of Z and current capital K at B = 0, current L = 4.1 (multiple entries preserved in source).
  - Table III: Average liquidity coverage ratio ((δL^* + π(L^*) Z_d − τ(y_min) + B^*(1 + r_f))/(D′(1 + r_d) − D_d)) reported as function of Z and K (multiple entries preserved in source).
  - Table IV (impact of bank regulation, medians):
    - Unregulated base: Loan (book) = 4.78; Net Bond Holdings (book) = −3.48; Bank Capital (book) = −0.70; Equity (mkt) = 4.49; Deposits (mkt) = 1.89; Enterprise Value (mkt) = 9.88; Government Value (mkt) = 0.54; Social value (mkt) = 10.56; Default/Closure Rate (pct) = 5.34.
    - Capital (k = 12% column example): Loan (book) = 6.37; Net Bond Holdings (book) = −3.89; Bank Capital (book) = 0.47; Equity (mkt) = 4.87; Enterprise Value (mkt) = 10.71; Government Value (mkt) = 0.88; Social value (mkt) = 11.72; Default/Closure Rate (pct) = 0.00.
    - Capital & liquidity (base k=4%, `=20% column example): Loan (book) = 2.68; Net Bond Holdings (book) = 0.16; Bank Capital (book) = 0.82; Equity (mkt) = 3.69; Enterprise Value (mkt) = 5.43; Government Value (mkt) = 0.37; Social value (mkt) = 5.91; Default/Closure Rate (pct) = 0.00.
  - Table V (taxation scenarios) and Table VI (role of equity issuance costs, adjustment costs, maturity transformation) report medians under varied parameter changes; entries preserved in source tables.
  - Table VII (bank closure rule comparison, medians):
    - Unregulated: Loan (book) = 4.78; Net Bond Holdings (book) = −3.48; Bank Capital (book) = −0.70; Equity (mkt) = 4.49; Enterprise Value (mkt) = 9.88; Government Value (mkt) = 0.54; Social value (mkt) = 10.56; Default/Closure Rate (pct) = 5.34.
    - k = 0 (nonnegative book equity): Loan (book) = 6.14; Net Bond Holdings (book) = −3.82; Bank Capital (book) = 0.31; Equity (mkt) = 4.82; Enterprise Value (mkt) = 10.58; Government Value (mkt) = 0.86; Social value (mkt) = 11.56; Default/Closure Rate (pct) = 21.75.
    - Closure rule: Loan (book) = 5.60; Net Bond Holdings (book) = −3.37; Bank Capital (book) = 0.25; Equity (mkt) = 5.48; Enterprise Value (mkt) = 10.76; Government Value (mkt) = 0.76; Social value (mkt) = 11.63; Default/Closure Rate (pct) = 21.75.

### Policy implications and interpretation (from results)
- Closure rule vs. capital requirements:
  - Closure rule can emulate effects of nonnegative book equity requirement (k = 0) under assumptions of zero government restructuring costs (γ = 0).
  - Closure rule increases loans, enterprise value, and social value relative to unregulated case when γ = 0 because the government captures positive equity value and avoids direct bankruptcy costs.
  - High rate of closure/reopening under the rule implies potentially higher real-world costs if γ > 0; thus closure rule may be more costly than capital requirements under positive reorganization costs.
- Regulatory tradeoffs:
  - Capital requirements increase capital ratios and reduce default/closure rates but can reduce enterprise and social value relative to unregulated benchmarks depending on parameter values.
  - Combining capital and liquidity requirements materially alters banks’ asset and liability composition (e.g., lower loan book in many calibrated cases) and changes enterprise and social values.
- Role of taxes, issuance costs, adjustment costs, and maturity transformation:
  - Changes in τ_+, τ_−, τ_B, λ, m_−, δ affect optimal policies and value metrics; specific quantitative effects documented in Tables V and VI.

*Source: _wp1272 - conclusion, the feasible set of the bank can be assumed to be [0,L_u]×[B_d,B_u].*

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