## _wp14237 - Section VI. we present our main results, and we make some final remarks in Section VII..

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---

### Methodology
- Two-stage simulation procedure:
  - Initial conditions use each bank’s measured portfolio (large and small credit exposures to real-economy and interbank borrowers) and liabilities composed of capital, interbank debt and deposits; depositors and other creditors are senior to interbank creditors.
  - Capital either:
    - set to a benchmark case based solely on portfolio loss distributions, or
    - allocated according to other allocations that partly rely on network measures.
  - Stage 1: simulate correlated exogenous shocks to all banks’ portfolios (returns on individual large loans and aggregated small loans) to generate correlated portfolio losses and initial (fundamental) defaults.
  - Stage 2: apply an extended version (Rogers and Veraart, 2013) of the fictitious contagion algorithm (Eisenberg and Noe, 2001), augmented with bankruptcy costs and a macroeconomic proxy for fire sales, producing contagious defaults.
  - Repeat stages for different capital allocations; optimization and incorporation of network interconnectedness measures are discussed in Section V. and Section B. respectively.

### Credit risk model (large and small loans)
- Large loans:
  - One-period CreditMetrics-style model calibrated to a one-year time span.
  - Systematic latent vector Y ∼ N(0, Σ) where Σ is a correlation matrix; intra-sector asset correlation ρ common to all sectors (value used in simulations: 0.20).
  - Asset return for borrower (j,k): X_{j,k} = √ρ Y_j + √(1−ρ) Z_{j,k}, with Z_{j,k} ∼ N(0,1).
  - 16 S&P rating classes plus aggregated “junk” CCC–C and default (D) treated as rating 18; ratings relabeled 1 (AAA) to 18 (default).
  - Migration thresholds θ set to match one-year transition probabilities p(R0,R1) (use 1981–2010 average one-year transition matrix for a global set of corporates).
  - Uniform maturity T = 4 years (closest digit to mean maturity 3.66 from Bundesbank borrower statistics).
  - Default-free interest rate r_f = 2%.
  - Loss-given-default LGD drawn from a beta distribution with expectation 0.39 and standard deviation 0.34.
  - Loan return formula: ret(R0,R1) = −1 + { D(C(R0),R1,1,T) + C(R0) if R1 < 18; (1 + C(R0)) (1 − LGD) if R1 = 18 }.
  - Each bank i’s euro return on large loans: ret_large,i = ∑_{j,k} LL_{i,j,k} ret(R_{j,k,0}, R_{j,k,1}), allowing synchronous defaults for shared borrowers.
- Small loans:
  - Exposures below the C1.5 mn reporting threshold are known only as a sector lump-sum and modeled portfolio-wise.
  - Use conditional central limit theorem: conditional on sector systematic factor Y_j, total losses approximated by a normal variable with mean and variance functions of Y_j.
  - Conditional variances calibrated using portfolio Hirschman-Herfindahl Index (HHI) forecasts derived from a large sample of small loans from a German commercial bank; estimates are sector specific.
  - Resulting euro return on each bank’s small loans denoted ret_small,i.

### Centrality measures and interbank network
- Interbank matrix X with entries x_{ij} = liability amount of bank i to bank j; adjacency matrix A where a_{ij} = 1 if x_{ij} > 0.
- Network characteristics:
  - Network size: 1764 nodes; number of links: 22,752; density: 0.7% (total possible links in directed network: 1764 × 1763 = 3,109,932).
- Centrality metrics used:
  - Out degree k_i = ∑_j a_{ij} (number of banks i borrows from).
  - In degree (number banks i lends to); Degree = Out degree + In degree.
  - Strength s_i = ∑_j x_{ij} (total interbank liabilities of a node); also compute interbank assets (inbound strength).
  - Opsahl centrality: OC_i = k_i^{(1−φ)} × s_i^{φ} with φ = 0.5.
  - Closeness centrality (Dangalchev, 2006): C_i = ∑_{j:j≠i} 2^{−d_{ij}} where d_{ij} is directed distance (∞ if no path).
  - Eigenvector centrality (Bonacich, 1987) from adjacency matrix A (normalized principal eigenvector v where Av = κ_1 v).
  - Weighted eigenvector centrality from liabilities matrix X.
  - Weighted normalized eigenvector from row-normalized X.
  - Local and global clustering coefficients (Watts and Strogatz, 1998).
  - Betweenness centrality based on geodesic paths.
  - Total assets used as a “centrality measure”.
  - First and second principal components of normalized centrality measures.
- Empirical network features:
  - Tiered interbank structure: about 20 banks (≈1%) lend to more than 100 banks; 30 banks borrow from at least 100 banks.
  - Strength statistics: 158 banks have total IB borrowed amount > C1bn; 27 banks have total interbank liabilities > C10bn. On assets side: 103 banks lend more than C1bn; 25 banks have interbank assets > C10bn.

### Data sources and aggregates
- Large-Exposures Database (LED) from Deutsche Bundesbank used at end of Q1 2011:
  - Interbank market consists of 1764 active lenders.
  - Around 400,000 credit exposures to more than 163,000 borrower units.
  - Borrowers in LED assigned to 100 fine-grained sectors; aggregated to EUROSTOXX’s 19 industry sectors plus Households and Public Sector for RM sectors (21 RM sectors total).
  - PD quantiles and means by RM sector documented; TOTAL exposures: 388,327 borrowers, Volume weight 100.0%, mean PD 1.5%.
- Borrower Statistics (BS) and Balance Sheet Statistics augment small-loan coverage; mapping from BS sectors to RM sectors via linear mapping.
- Market data: Merill Lynch euro corporate spreads (ER10, ER20, ER30, ER40, HE10, HE20, HE30) from April 1999 to June 2011; asset correlations from EUROSTOXX weekly returns April 2006 – March 2011; S&P migration matrix (Standard and Poor’s, 2011).
- German banking system (Q1 2011 totals):
  - 1921 MFIs registered; total balance sheet C8,233 bn.
  - Interbank network of 1764 active banks holds total assets C7,791 bn; 77% represent large loans and 23% small loans.

### Modeling contagion and bankruptcy costs
- Loss definitions:
  - Fundamental losses L_fund,i ≡ −(ret_large,i + ret_small,i).
  - Total portfolio losses L_i = L_fund,i + L_IB,i.
  - Bank default indicator D_i = 1 if K_i < L_i, 0 otherwise.
- Interbank contagion allocation:
  - Interbank liabilities row sums l_i = ∑_{j ≠ i} x_{ij}.
  - Proportionality matrix π_{ij} = x_{ij} / l_i if l_i > 0; 0 otherwise.
  - Loss on liability side from default of i: Λ_{IB,i} = min(l_i, max(0, L_i + BC_i D_i − K_i)), where BC_i are bankruptcy costs and D_i indicates default.
  - Interbank asset losses: L_IB = Π^> Λ_{IB} (vector form).
  - Fixed-point iteration using monotone operator Φ (Rogers and Veraart, 2013) converges to unique minimal-loss fixed point L_∞.
- Bankruptcy costs (BCs) modeled as:
  - BC_i ≡ φ (TotalAssets_i − L_fund,i) + λ(L_fund) max(0, L_fund,i),
  - where φ is proportion of assets lost due to litigation and legal costs (analysis sets φ = 5%); λ(L_fund) is a monotonic function of total fundamental losses L_fund ≡ ∑_i max(0, L_fund,i − K_i), chosen as the cumulative distribution function of L_fund.
- Primary target function for optimization:
  - Expected BCs: EBC = E ∑_i BC_i D_i (Equation (8)).
- Alternative target functions (expected total loss to equity or non-bank debt holders) discussed and shown to be equivalent to expected BCs in this setup.

### Optimization of capital allocation
- Benchmark capital: VaR on portfolio at α = 99.9% (in line with Basel II banking book level), denoted K_{α,i}; interbank loans treated as ordinary loans in this benchmark (merged into portfolio sector no. 17).
- Total required capital in benchmark TK_α ≡ ∑_i K_{α,i} is held constant across reallocations.
- Simple centrality-based capital rule:
  - K_simple,i(β) ≡ K_{α,i} (1 − β + β a C_i), where C_i is chosen centrality measure, a chosen so ∑_j K_simple,j(β) = TK_α.
  - Parameter β denotes fraction of benchmark capital redistributed toward centrality-based rule.
- Minimum capital floor:
  - Each bank must hold at least K_{min,i} = bank i’s VaR at α = 99%.
  - Final capital rule: K_centr,i(β) ≡ max(K_{min,i}, K_{α,i} [1 − β + β τ(β) a C_i]), where tuning factor τ(β) numerically reestablishes TK_α if floor binds.
  - Tuning factor τ(β) found to be virtually 1 (0.999 < τ(β) ≤ 1) even up to β = 30%; at most 14 banks have the floor binding.
- Optimization over β and choice of single centrality measure; bivariate optimizations over combinations of centrality measures explored but improvements negligible.

### Results — key findings and statistics
- Benchmark point corresponds to β = 0 (purely VaR(α = 99.9%) based).
- Performance of single-measure reallocations (Expected BCs in T-EUR; Saving relative to benchmark 1,008,000 T-EUR; Optimal beta):
  - Adjacency Eigenvector: Expected BCs: 861,000; Saving: 14.6 percent; Optimal beta: 12 percent
  - Eigenvector (weighted normalized): Expected BCs: 898,000; Saving: 10.9 percent; Optimal beta: 8 percent
  - Closeness: Expected BCs: 900,000; Saving: 10.7 percent; Optimal beta: 24 percent
  - Opsahl Centrality: Expected BCs: 910,000; Saving: 9.8 percent; Optimal beta: 8 percent
  - Out Degree: Expected BCs: 912,000; Saving: 9.6 percent; Optimal beta: 8 percent
  - IB Liabilities: Expected BCs: 917,000; Saving: 9.18 percent; Optimal beta: 8 percent
  - Degree: Expected BCs: 920,000; Saving: 8.7 percent; Optimal beta: 10 percent
  - Eigenvector (weighted): Expected BCs: 928,000; Saving: 8.06 percent; Optimal beta: 8 percent
  - 1st Principal Component: Expected BCs: 946,000; Saving: 6.24 percent; Optimal beta: 6 percent
  - Weighted Betweenness: Expected BCs: 991,000; Saving: 1.84 percent; Optimal beta: 2 percent
  - In Degree: Expected BCs: 995,000; Saving: 1.32 percent; Optimal beta: 2 percent
  - Total Assets: Expected BCs: 998,000; Saving: 1.02 percent; Optimal beta: 2 percent
  - IB Assets, Clustering, 2nd Principal Component: Expected BCs: 1,008,000; Saving: 0.00 percent (dominated; no improvement)
  - Benchmark (purely VaR based): Expected BCs: 1,008,000.
- Main patterns:
  - Some centrality measures are dominated (Clustering, IB Assets, 2nd Principal Component).
  - Best improvement achieved by Adjacency Eigenvector: reallocating 12% of capital yields a 14.6 percent reduction in Expected BCs.
  - Closeness also effective (10.7 percent saving) but requires higher β (24%) at its optimum.
  - Several other measures (Opsahl, weighted eigenvectors, Out Degree, Degree) reduce Expected BCs by about 8–10 percent with β ≈ 8–10%.
- Decomposition of Expected BCs:
  - Capital reallocations based on centrality measures tend to increase fundamental defaults (pre-contagion) for many (presumably smaller) banks while reducing contagious defaults and total bankruptcy costs.
  - For Closeness, fundamental defaults mimic benchmark across β and decline in contagion losses dominates, leading to lower total losses with little pre-contagion cost increase until β ≈ 15% inflection.
  - Adjacency Eigenvector reduces post-contagion losses sharply at small expense in fundamental losses until an inflection near β = 15% where fundamental losses rise steeply.
- Distributional effects:
  - Optimal reallocations spread the distribution of bank PDs before contagion to the right (more smaller-bank PDs increase) while reducing PDs of some high-impact banks post-contagion.
  - Post-contagion PDs are widely distributed and on average 35 percent higher than the nominal 0.1% label implied by a VaR(α = 99.9%) measure treating interbank loans as ordinary loans.
- Tail performance (systemic meltdowns):
  - Both Closeness and Adjacency Eigenvector outperform benchmark for catastrophic meltdowns in Expected BCs frequency density tails (BCs > C100 billion).
  - Adjacency Eigenvector dominates Closeness in most of the tail modes; for extreme BCs > C780 billion, all rules perform similarly poorly.
- Combining centrality measures:
  - Combining two measures yields only marginal improvement (0.2% in expected BCs) due to lack of orthogonality in their impacts; optimal solutions tended to put full weight on a single measure.
- Relative changes in required capital across banks:
  - Most banks experience a cut in required capital under centrality-based reallocations; Adjacency Eigenvector reduces capital most drastically for the majority by 12% of benchmark capital; Closeness reduces by about 5% for the majority.
  - Reallocations are generally “mild” for the majority of banks, concentrating extra capital on a small set of highly central banks.

### Policy-relevant inferences and recommendations
- Capital requirements that incorporate network centrality measures can reduce expected total bankruptcy costs versus a benchmark VaR(α = 99.9%) allocation.
- Adjacency Eigenvector centrality provides the largest reduction in Expected BCs (14.6% saving at β = 12%).
- Moderately reallocating capital (β around 8–12% for several effective measures) away from many small banks towards a few systemically important banks can substantially reduce post-contagion losses, at the cost of raising fundamental PDs for a larger set of smaller banks.
- Simple, transparent, and easily computable rules are preferred for political feasibility and to limit discretion and lobbying; the paper focuses on simple single-measure reallocations for this reason.
- Minimum capital floors (K_{min,i} = VaR at α = 99%) and global capital neutrality (keeping TK_α constant) are practical constraints in designing reallocations.
- Combining centrality measures yields little additional benefit over the best single centrality measure given observed lack of orthogonality.
- Further considerations for policy adoption:
  - Implementation challenges include legal, political, and endogenous network responses (banks will adjust behavior in response to capital rules).
  - Possible extensions include bailout funding designs, insurance premia based on centrality, and inclusion of other systemically important institutions with additional reporting.

### Conclusion
- The framework combining detailed credit portfolio simulations, confidential bilateral lending data, and network-based centrality measures allows analysis of capital allocation effects on systemic stability.
- Network-aware capital reallocations outperform a portfolio-only VaR benchmark in reducing expected bankruptcy costs.
- Adjacency Eigenvector centrality based capital reallocation (12% of capital reallocated) achieved the largest reduction in expected system losses (14.6%).
- Results support the view that systemic capital requirements should incorporate interconnectedness measures that account for both individual bank centrality and the importance of their neighbors.
- Future research directions include endogenous network formation, bailout rule design based on centrality, and extending methods to other financial sectors.

*Source — _wp14237 - Section VI. we present our main results, and we make some final remarks in Section VII.. (PDF chapter/section).*

### Section VI. we present our main results, and we make some final remarks in Section VII..

### _wp14237 - Section VI. we present our main results, and we make some final remarks in Section VII..

### Methodology
- Two-stage simulation procedure with an initial condition using each bank’s measured portfolio (large and small credit exposures to real-economy and interbank borrowers), and liabilities composed of capital, interbank debt and deposits; depositors and other creditors are senior to interbank creditors.
- Capital is either:
  - set to a benchmark case based solely on portfolio loss distributions, or
  - allocated according to other allocations that partly rely on network measures.
- Stage 1: simulate correlated exogenous shocks to all banks’ portfolios (returns on individual large loans and aggregated small loans) to generate correlated portfolio losses and initial (fundamental) defaults.
- Stage 2: apply an extended version (Rogers and Veraart, 2013) of the fictitious contagion algorithm (Eisenberg and Noe, 2001), augmented with bankruptcy costs and a macroeconomic proxy for fire sales, producing contagious defaults.
- Repeat stages for different capital allocations; optimization and incorporation of network interconnectedness measures are discussed in Section V. and Section B. respectively.

### Credit risk model (large and small loans)
- Large loans:
  - One-period CreditMetrics-style model calibrated to a one-year time span.
  - Systematic latent vector Y ∼ N(0, Σ) where Σ is a correlation matrix; intra-sector asset correlation ρ common to all sectors (value used in simulations: 0.20).
  - Asset return for borrower (j,k): X_{j,k} = √ρ Y_j + √(1−ρ) Z_{j,k}, with Z_{j,k} ∼ N(0,1).
  - 16 S&P rating classes plus aggregated “junk” CCC–C and default (D) treated as rating 18; ratings relabeled 1 (AAA) to 18 (default).
  - Migration thresholds θ set to match one-year transition probabilities p(R0,R1) (use 1981–2010 average one-year transition matrix for a global set of corporates).
  - Uniform maturity T = 4 years (closest digit to mean maturity 3.66 from Bundesbank borrower statistics).
  - Default-free interest rate r_f = 2%.
  - Loss-given-default LGD drawn from a beta distribution with expectation 0.39 and standard deviation 0.34.
  - Loan return formula: ret(R0,R1) = −1 + { D(C(R0),R1,1,T) + C(R0) if R1 < 18; (1 + C(R0)) (1 − LGD) if R1 = 18 }.
  - Each bank i’s euro return on large loans: ret_large,i = ∑_{j,k} LL_{i,j,k} ret(R_{j,k,0}, R_{j,k,1}), allowing synchronous defaults for shared borrowers.
- Small loans:
  - Exposures below the C1.5 mn reporting threshold are known only as a sector lump-sum and modeled portfolio-wise.
  - Use conditional central limit theorem: conditional on sector systematic factor Y_j, total losses approximated by a normal variable with mean and variance functions of Y_j.
  - Conditional variances calibrated using portfolio Hirschman-Herfindahl Index (HHI) forecasts derived from a large sample of small loans from a German commercial bank; estimates are sector specific.
  - Resulting euro return on each bank’s small loans denoted ret_small,i.

### Centrality measures and interbank network
- Interbank matrix X with entries x_{ij} = liability amount of bank i to bank j; adjacency matrix A where a_{ij} = 1 if x_{ij} > 0.
- Network characteristics:
  - Network size: 1764 nodes; number of links: 22,752; density: 0.7% (total possible links in directed network: 1764 × 1763 = 3,109,932).
- Centrality metrics used:
  - Out degree k_i = ∑_j a_{ij} (number of banks i borrows from).
  - In degree (number banks i lends to); Degree = Out degree + In degree.
  - Strength s_i = ∑_j x_{ij} (total interbank liabilities of a node); also compute interbank assets (inbound strength).
  - Opsahl centrality: OC_i = k_i^{(1−φ)} × s_i^{φ} with φ = 0.5 (geometric mean of strength and degree).
  - Closeness centrality (Dangalchev, 2006): C_i = ∑_{j:j≠i} 2^{−d_{ij}} where d_{ij} is directed distance (∞ if no path).
  - Eigenvector centrality (Bonacich, 1987) from adjacency matrix A (normalized principal eigenvector v where Av = κ_1 v).
  - Weighted eigenvector centrality from liabilities matrix X.
  - Weighted normalized eigenvector from row-normalized X.
  - Local and global clustering coefficients (Watts and Strogatz, 1998).
  - Betweenness centrality based on geodesic paths.
  - Total assets used as a “centrality measure”.
  - First and second principal components of normalized centrality measures.
- Empirical network features:
  - Tiered interbank structure: about 20 banks (≈1%) lend to more than 100 banks; 30 banks borrow from at least 100 banks.
  - Strength statistics: 158 banks have total IB borrowed amount > C1bn; 27 banks have total interbank liabilities > C10bn. On assets side: 103 banks lend more than C1bn; 25 banks have interbank assets > C10bn.

### Data sources and aggregates
- Large-Exposures Database (LED) from Deutsche Bundesbank used at end of Q1 2011:
  - Interbank market consists of 1764 active lenders.
  - Around 400,000 credit exposures to more than 163,000 borrower units.
  - Borrowers in LED assigned to 100 fine-grained sectors; aggregated to EUROSTOXX’s 19 industry sectors plus Households and Public Sector for RM sectors (21 RM sectors total).
  - PD quantiles and means by RM sector documented (see Table 1 in source); TOTAL exposures: 388,327 borrowers, Volume weight 100.0%, mean PD 1.5% (table values preserved in source).
- Borrower Statistics (BS) and Balance Sheet Statistics augment small-loan coverage; mapping from BS sectors to RM sectors via linear mapping.
- Market data: Merill Lynch euro corporate spreads (ER10, ER20, ER30, ER40, HE10, HE20, HE30) from April 1999 to June 2011; asset correlations from EUROSTOXX weekly returns April 2006 – March 2011; S&P migration matrix (Standard and Poor’s, 2011).
- German banking system (Q1 2011 totals):
  - 1921 MFIs registered; total balance sheet C8,233 bn.
  - Interbank network of 1764 active banks holds total assets C7,791 bn; 77% represent large loans and 23% small loans.

### Modeling contagion and bankruptcy costs
- Loss definitions:
  - Fundamental losses L_fund,i ≡ −(ret_large,i + ret_small,i).
  - Total portfolio losses L_i = L_fund,i + L_IB,i.
  - Bank default indicator D_i = 1 if K_i < L_i, 0 otherwise.
- Interbank contagion allocation:
  - Interbank liabilities row sums l_i = ∑_{j ≠ i} x_{ij}.
  - Proportionality matrix π_{ij} = x_{ij} / l_i if l_i > 0; 0 otherwise.
  - Loss on liability side from default of i: Λ_{IB,i} = min(l_i, max(0, L_i + BC_i D_i − K_i)), where BC_i are bankruptcy costs and D_i indicates default.
  - Interbank asset losses: L_IB = Π^> Λ_{IB} (vector form).
  - Fixed-point iteration using monotone operator Φ (Rogers and Veraart, 2013) converges to unique minimal-loss fixed point L_∞.
- Bankruptcy costs (BCs) modeled as:
  - BC_i ≡ φ (TotalAssets_i − L_fund,i) + λ(L_fund) max(0, L_fund,i),
  - where φ is proportion of assets lost due to litigation and legal costs (analysis sets φ = 5%); λ(L_fund) is a monotonic function of total fundamental losses L_fund ≡ ∑_i max(0, L_fund,i − K_i), chosen as the cumulative distribution function of L_fund.
- Primary target function for optimization:
  - Expected BCs: EBC = E ∑_i BC_i D_i (Equation (8)).
- Alternative target functions (expected total loss to equity or non-bank debt holders) discussed and shown to be equivalent to expected BCs in this setup.

### Optimization of capital allocation
- Benchmark capital: VaR on portfolio at α = 99.9% (in line with Basel II banking book level), denoted K_{α,i}; interbank loans treated as ordinary loans in this benchmark (merged into portfolio sector no. 17).
- Total required capital in benchmark TK_α ≡ ∑_i K_{α,i} is held constant across reallocations.
- Simple centrality-based capital rule:
  - K_simple,i(β) ≡ K_{α,i} (1 − β + β a C_i), where C_i is chosen centrality measure, a chosen so ∑_j K_simple,j(β) = TK_α.
  - Parameter β denotes fraction of benchmark capital redistributed toward centrality-based rule.
- Minimum capital floor:
  - Each bank must hold at least K_{min,i} = bank i’s VaR at α = 99%.
  - Final capital rule: K_centr,i(β) ≡ max(K_{min,i}, K_{α,i} [1 − β + β τ(β) a C_i]), where tuning factor τ(β) numerically reestablishes TK_α if floor binds.
  - Tuning factor τ(β) found to be virtually 1 (0.999 < τ(β) ≤ 1) even up to β = 30%; at most 14 banks have the floor binding.
- Optimization over β and choice of single centrality measure; bivariate optimizations over combinations of centrality measures explored but improvements negligible.

### Results — key findings and statistics
- Benchmark point corresponds to β = 0 (purely VaR(α = 99.9%) based).
- Optimal reallocations and performance (Table 3 summary):
  - Adjacency Eigenvector (adjacency eigenvector centrality):
    - Expected BCs: 861,000 (T-EUR)
    - Saving: 14.6 percent (relative to benchmark Expected BCs 1,008,000 T-EUR)
    - Optimal beta: 12 percent
  - Eigenvector (weighted normalized):
    - Expected BCs: 898,000 (T-EUR); Saving: 10.9 percent; Optimal beta: 8 percent
  - Closeness:
    - Expected BCs: 900,000 (T-EUR); Saving: 10.7 percent; Optimal beta: 24 percent
  - Opsahl Centrality:
    - Expected BCs: 910,000 (T-EUR); Saving: 9.8 percent; Optimal beta: 8 percent
  - Out Degree:
    - Expected BCs: 912,000 (T-EUR); Saving: 9.6 percent; Optimal beta: 8 percent
  - IB Liabilities:
    - Expected BCs: 917,000 (T-EUR); Saving: 9.18 percent; Optimal beta: 8 percent
  - Degree:
    - Expected BCs: 920,000 (T-EUR); Saving: 8.7 percent; Optimal beta: 10 percent
  - Eigenvector (weighted):
    - Expected BCs: 928,000 (T-EUR); Saving: 8.06 percent; Optimal beta: 8 percent
  - 1st Principal Component:
    - Expected BCs: 946,000 (T-EUR); Saving: 6.24 percent; Optimal beta: 6 percent
  - Weighted Betweenness:
    - Expected BCs: 991,000 (T-EUR); Saving: 1.84 percent; Optimal beta: 2 percent
  - In Degree:
    - Expected BCs: 995,000 (T-EUR); Saving: 1.32 percent; Optimal beta: 2 percent
  - Total Assets:
    - Expected BCs: 998,000 (T-EUR); Saving: 1.02 percent; Optimal beta: 2 percent
  - IB Assets, Clustering, 2nd Principal Component:
    - Expected BCs: 1,008,000 (T-EUR); Saving: 0.00 percent (dominated; no improvement over benchmark)
  - Benchmark (purely VaR based): Expected BCs: 1,008,000 (T-EUR).
- Main patterns:
  - Some centrality measures are dominated (Clustering, IB Assets, 2nd Principal Component).
  - Best improvement achieved by Adjacency Eigenvector: reallocating 12% of capital yields a 14.6 percent reduction in Expected BCs.
  - Closeness also effective (10.7 percent saving) but requires higher β (24%) at its optimum.
  - Several other measures (Opsahl, weighted eigenvectors, Out Degree, Degree) reduce Expected BCs by about 8–10 percent with β ≈ 8–10%.
- Decomposition of Expected BCs:
  - Capital reallocations based on centrality measures tend to increase fundamental defaults (pre-contagion) for many (presumably smaller) banks while reducing contagious defaults and total bankruptcy costs.
  - For Closeness, fundamental defaults mimic benchmark across β and decline in contagion losses dominates, leading to lower total losses with little pre-contagion cost increase until β ≈ 15% inflection.
  - Adjacency Eigenvector reduces post-contagion losses sharply at small expense in fundamental losses until an inflection near β = 15% where fundamental losses rise steeply.
- Distributional effects:
  - Optimal reallocations spread the distribution of bank PDs before contagion to the right (more smaller-bank PDs increase) while reducing PDs of some high-impact banks post-contagion.
  - Post-contagion PDs are widely distributed and on average 35 percent higher than the nominal 0.1% label implied by a VaR(α = 99.9%) measure treating interbank loans as ordinary loans.
- Tail performance (systemic meltdowns):
  - Both Closeness and Adjacency Eigenvector outperform benchmark for catastrophic meltdowns in Expected BCs frequency density tails (BCs > C100 billion).
  - Adjacency Eigenvector dominates Closeness in most of the tail modes; for extreme BCs > C780 billion, all rules perform similarly poorly.
- Combining centrality measures:
  - Combining two measures yields only marginal improvement (0.2% in expected BCs) due to lack of orthogonality in their impacts; optimal solutions tended to put full weight on a single measure.
- Relative changes in required capital across banks:
  - Most banks experience a cut in required capital under centrality-based reallocations; Adjacency Eigenvector reduces capital most drastically for the majority by 12% of benchmark capital; Closeness reduces by about 5% for the majority.
  - Reallocations are generally “mild” for the majority of banks, concentrating extra capital on a small set of highly central banks.

### Policy-relevant inferences and recommendations
- Capital requirements that incorporate network centrality measures can reduce expected total bankruptcy costs versus a benchmark VaR(α = 99.9%) allocation.
- Adjacency Eigenvector centrality provides the largest reduction in Expected BCs (14.6% saving at β = 12%).
- Moderately reallocating capital (β around 8–12% for several effective measures) away from many small banks towards a few systemically important banks can substantially reduce post-contagion losses, at the cost of raising fundamental PDs for a larger set of smaller banks.
- Simple, transparent, and easily computable rules are preferred for political feasibility and to limit discretion and lobbying; the paper focuses on simple single-measure reallocations for this reason.
- Minimum capital floors (K_{min,i} = VaR at α = 99%) and global capital neutrality (keeping TK_α constant) are practical constraints in designing reallocations.
- Combining centrality measures yields little additional benefit over the best single centrality measure given observed lack of orthogonality.
- Further considerations for policy adoption:
  - Implementation challenges include legal, political, and endogenous network responses (banks will adjust behavior in response to capital rules).
  - Extensions could include bailout funding designs, insurance premia based on centrality, and inclusion of other systemically important institutions with additional reporting.

### Conclusion (summary)
- The framework combining detailed credit portfolio simulations, confidential bilateral lending data, and network-based centrality measures allows analysis of capital allocation effects on systemic stability.
- Network-aware capital reallocations outperform a portfolio-only VaR benchmark in reducing expected bankruptcy costs.
- Adjacency Eigenvector centrality based capital reallocation (12% of capital reallocated) achieved the largest reduction in expected system losses (14.6%).
- Results support the view that systemic capital requirements should incorporate interconnectedness measures that account for both individual bank centrality and the importance of their neighbors.
- Future research directions include endogenous network formation, bailout rule design based on centrality, and extending methods to other financial sectors.

*Italic: Source — _wp14237 - Section VI. we present our main results, and we make some final remarks in Section VII.. (PDF chapter/section).*

### References

### References

### Network structure, systemic risk, and contagion
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### Measures, centrality, and systemic-importance algorithms
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### Modeling, simulation, and robustness of systemic-risk measures
- Alessandri, P., P. Gai, S. Kapadia, N. Mora, and C. Puhr (2009). Towards a framework for quantifying systemic stability.International Journal of Central Banking September, 47–81.
- Cont, R., A. Moussa, and E. B. e Santos (2013).Handbook of Systemic Risk, Chapter Network structure and systemic risk in banking systems, pp. 327–368. Cambridge University Press. Mimeo.
- Gai, P. and S. Kapadia (2010). Contagion in financial networks.Proceedings of the Royal Society, Series A: Mathematical, Physical and Engineering Sciences 466, 2401–2423.
- Löffler, G. and P. Raupach (2013). Robustness and informativeness of systemic risk measures. Deutsche Bundesbank Discussion Paper No. 04/2013.
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### Empirical credit-risk, recovery rates, and databases
- Altman, E. (1984). A further empirical investigation of the bankruptcy cost question.The Jornal of Finance 39, 1067 – 1089.
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- Schmieder, C. (2006). The Deutsche Bundesbank’s large credit database (BAKIS-M and MiMiK). Schmollers Jahrbuch 126, 653–663.

### Macroprudential policy, regulatory, and speech literature
- Basel Committee on Banking Supervision (2011). Global systemically important banks: Assessment methodology and the additional loss absorbency requirement. Bank for International Settlements.
- Gauthier, C., A. Lehar, and M. Souissi (2012). Macroprudential capital requirements and systemic risk.Journal of Financial Intermediation 21, 594–618.
- Haldane, A. (2009). Rethinking the financial network. Speech delivered at the Financial Student Association, Amsterdam.
- Standard and Poor’s (2011). 2010 annual global corporate default study and rating transitions.
- Yellen, J. L. (2013). Interconnectedness and systemic risk: Lessons from the financial crisis and policy implications. Speech at the American Economic Association/American Finance Association Joint Luncheon, San Diego, California.

### Foundational textbooks and methodological references
- Newman, M. (2010).Networks: An Introduction. Oxford, UK: Oxford University Press.

*References as listed in the source PDF.*

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