## 1fraea2019007

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### Executive summary and model purpose
- Macroprudential policy challenge:
  - Identifying growth of financial and macroeconomic variables above and below potential; gaps are unobserved (1).
- Model presented:
  - Captures (i) sectoral dynamics of firms and banks and feedbacks between them, (ii) capital and default risk dynamics of each sector, (iii) capital and risk gaps (deviations from the welfare maximizing optimum), and (iv) provides a quantitative method for measurement (2).
- Definition of potential:
  - Potential levels of capital and default risk are those a central planner would choose where corporate and bank capital levels are flexible (3).
- Use for policy:
  - Framework is silent on implementation instruments but can be used as a signaling device to indicate when macroprudential action is necessary; any policy that closes capital and risk gaps considering general equilibrium effects can be evaluated with the model (4).
- Data and estimation:
  - Model is taken from French aggregate publicly available data and estimated with Bayesian estimation techniques (5).

### Key empirical findings on default risk and capital
- Default risk dynamics and drivers:
  - Default risk fluctuates between being too high and too low; risk is too high in four episodes: prior to the Technology Crisis, prior to the Global Financial Crisis, prior to the Sovereign Debt Crisis, and now (6).
  - Corporate default risk gap was particularly elevated during the Technology Crisis and now, although default risk is now relatively low in absolute terms (6).
  - Banking sector default risk was particularly elevated during the Global Financial Crisis and appears now adequate (6).
  - Corporate default risk is driven significantly by cross-sectional, idiosyncratic risk (‘eF’ — yellow bars) and by productivity shocks (‘ey’ — blue bars).
  - Banking default risk is mostly driven by aggregate risk (‘eB’ — light blue bars).
  - Figure axis notes preserved:
    - For corporate PD figure: the y-axis denotes percentages (0.01 equals to 1 percent).
    - For banking PD figure: the y-axis denotes percentages (0.01 equals to 1 percent).
- Capital gaps:
  - Episodes where more corporate capital is needed (positive gaps): prior to the Technology Crisis, prior to the Global Financial Crisis, prior to the Sovereign Debt Crisis, and now.
  - Bank capital was too low during the Global Financial Crisis and the Sovereign Debt Crisis, has strengthened significantly, and appears now to be adequate (7).
  - Figure axis notes preserved:
    - Corporate capital gap figure: the y-axis denotes percentages (0.05 equals to a 5 percent gap relative to the total market cap).
    - Banking capital gap figure: the y-axis denotes percentages (0.2 equals to a 20 percent gap relative to the nominal amount of aggregate capital).
- Macro stress test (scenario and outcomes):
  - Scenario: structural shocks from 2008–2011 (Global Financial Crisis) applied to current economy to simulate 2018–2021.
  - Results:
    - Bank capital is about adequate under the stress scenario; more corporate capital is needed under adverse conditions.
    - Firms are more vulnerable than banks at the current juncture.
    - Under adverse conditions, the corporate risk gap reaches unprecedented highs; the bank risk gap is considerably lower than during the Global Financial Crisis even if stress intensifies.
    - Figure notes: for risk, 0.005 equals to 0.5 percent excessive default risk in absolute terms; for capital, 0.2 equals to a 20 percent gap relative to the nominal amount of aggregate capital.

### Model structure, frictions, and gap measurement
- Model structure:
  - Core: a standard real business cycle model enhanced with a financial system of three agent sets: financial intermediaries, financial investors, and nonfinancial firms (13).
  - Agents and flows:
    - Financial intermediaries borrow from financial investors and on-lend to nonfinancial firms; financial investors hold equity of firms and intermediaries (16–17).
  - Shocks:
    - Macro shocks: total factor productivity (‘ey’), government spending (‘eg’), investment efficiency (‘ei’), labor market tightening (‘en’) (15).
    - Financial shocks: idiosyncratic cross-sectional return-on-asset variance (‘eF’), aggregate return-on-asset variance (‘eB’), and financial intermediary income shock (‘ek’) (empirical strategy).
- Financial frictions and optimality:
  - Three financial frictions:
    - Tax treatment: income of nonfinancial firms and intermediaries is tax-deductible; income of financial investors is not (benefit of financial intermediation).
    - Defaults of nonfinancial corporates trigger a screening cost; optimal lending rate trades off intermediation benefit vs cost of default.
    - Defaults of financial intermediaries result in lower productivity; cost-of-funding trades off intermediation benefit vs productivity loss in case of default.
  - Nonfinancial firms maximize with respect to three choice variables: intermediaries’ funding (hence firms’ liabilities), lending rates, and intermediaries’ cost-of-funding.
- Capital gaps (concept and calculation):
  - Capital is sticky and accumulated out of retained earnings (as in Christiano, Motto and Rostagno (2014)).
  - Capital gap = welfare-maximizing optimal capital − own capital accumulated out of retained earnings.
  - The welfare-maximizing problem solves for five choice variables (including optimal capital for firms and intermediaries) versus three in the model solution; deviations of capital from potential imply deviations of risk from potential.
  - Default thresholds and constraints are embedded in the welfare maximization subject to funding constraints, FI participation and capital constraints, and wholesale FI debt investor participation constraints.

### Empirical strategy and French application
- Data and sample:
  - Quarterly macroeconomic and financial time series between 2000:Q4 and 2018:Q2.
  - Macroeconomic series: output, investment, employment, and consumption (all demeaned percentage changes).
  - Financial series: corporate credit spread (demeaned percentages), financial credit spread (demeaned percentages), and aggregate bank capital (demeaned percentage changes).
- Estimation approach:
  - Metropolis-Hastings algorithm (Bayesian estimation) used to find parameters that maximize likelihood; standard parameters calibrated.
  - Models with the presented frictions improve model fit relative to the standard RBC model.
- Outputs reported:
  - Deviations from potential (gaps) for:
    - Capital of nonfinancial firms
    - Capital of financial intermediaries
    - Default risk of nonfinancial firms
    - Default risk of financial intermediaries
  - Shock decomposition identifying historical realizations consistent with observed data.

### Interconnectedness, datasets, and main metrics
- Datasets exploited:
  - Unilateral (domestic and cross-border) exposures in marketable securities for banks, insurers, and funds.
  - Bilateral reporting dataset for domestic assets and liabilities between French financial entities.
  - BIS consolidated banking statistics to examine interbank loans at country aggregation level.
- Coverage and levels:
  - Unilateral dataset covers EUR 4.8 trillion in assets (of which EUR 1.1 trillion to French financial entities).
  - Bilateral dataset covers EUR 835 billion in assets between French financial entities.
  - Unilateral dataset coverage relative to total assets: banks 10.5 percent, insurers 92.7 percent, funds 96.3 percent.
  - Bilateral dataset coverage relative to total assets: banks 2.2 percent, insurers 29.8 percent, funds 25.7 percent.
- Cross-border exposures (2017 and changes since 2011):
  - Total marketable securities across banks, insurers, and funds: EUR 4.8 trillion, an increase of 27 percent since 2011.
  - Insurance companies: EUR 2.4 trillion (+36 percent since 2011).
  - Banks: EUR 896 billion (-12 percent since 2011).
  - Funds: EUR 1.5 trillion (+30 percent since 2011).
  - Insurers cross-border exposures: EUR 1.15 trillion (+25 percent since 2011); cross-border share 49 percent (-2 percent).
  - Funds cross-border exposures: EUR 784 billion (+47 percent since 2011); cross-border share 52 percent (+8 percent).
  - Banks cross-border holdings reduced to EUR 375 million in 2017 (-39 percent since 2011); cross-border share steady at 42 percent of banks’ marketable securities holdings since 2011.
- Geographic concentration:
  - Highest individual country exposure: Luxembourg: EUR 337 billion in marketable securities holdings.
- Instrument composition highlights:
  - Banks: debt securities comprise 80 percent of domestic positions and 69 percent of cross-border positions.
  - Insurance companies: debt securities comprise 76 percent of domestic and 72 percent of cross-border securities; domestic equity exposures increased since 2015 (Solvency II structural break noted).
  - Funds (domestic positions since 2012): debt 33 percent, equity 33 percent, fund shares 34 percent.
  - Funds (abroad): debt 55 percent, equity 33 percent, foreign fund shares 12 percent.

### Network analysis, common exposures, and contagion
- Network properties (bilateral dataset, 2017):
  - Network size: Funds 9,670; Banks 132; Insurance companies 130; Total vertices 9,932.
  - Total edges (exposures from a bank or insurance company to a fund): 61,348.
  - Density: 0.06 percent.
  - Average distance: 3.24 (varies between 1 and 14).
  - Degree distribution: 75 percent of entities connected to at most 10 others in 2017; three banks and six insurance companies have more than 1000 links.
- Common exposure methodology and significance:
  - Uses linear transformation of distance from portfolio correlations following Giudici, Sarlin and Spelta (2017).
  - t-statistic: t = r √(n−2) / √(1−r2). For 2026 degrees of freedom, one-sided test critical values 2.33, 1.96, and 1.28 correspond to out-flow common exposures (correlations) of 0.64 (0.07), 0.62 (0.05), and 0.61 (0.04) at the 1 percent, 5 percent, and 10 percent levels respectively.
  - For out-flow cross-border assets, significance is achieved at all conventional levels for every out-flow common exposure measure reported.
- Cross-border common exposure findings (funds and insurers):
  - Funds sector shows relatively high internal common exposure; reported pair indices:
    - Equity–Mixed: 1.32
    - Bond–Mixed: 1.24
    - Mixed–Other: 1.40
    - Equity–Other: 1.17
  - Insurance–bank and insurer–mixed common exposures:
    - Insurer–Bank: 1.20
    - Insurer–Mixed: 1.19
  - Bond funds have highly significant overlap with every other sector; common exposures for significant pairs have been increasing since 2015.
- Domestic common exposures (bilateral dataset) and clusters:
  - Significant domestic out-flow clusters:
    - Cluster 1: bond, other, and money market funds (example Bond–Other or Bond–MMF = 1.40).
    - Cluster 2: mixed funds, banks, and insurers (Banks–Insurers = 1.30; Insurers–Mixed = 1.52).
  - In-flow (liabilities) common exposures centered on mixed funds:
    - Mixed–Equity: 1.77
    - Mixed–Bond: 1.80
    - Mixed–Hedge: 1.82
    - Mixed–Other: 1.76
    - Mixed–MMF: 1.40
  - Statistical threshold: in bilateral dataset, a common exposure of 1.07 is significant at the 1 percent level (14 degrees of freedom critical values noted in source).
- Contagion analysis (interbank loans using BIS consolidated statistics):
  - Countries included: Austria, Australia, Belgium, Canada, Switzerland, Chile, Germany, Spain, Finland, France, Greece, the United Kingdom, Hong Kong SAR, Ireland, India, Italy, Japan, Korea, the Netherlands, Norway, Portugal, Sweden, Singapore, Turkey, and the United States.
  - Shock types:
    - Credit shock: loss-given-default λ = 100 percent on interbank loans of the defaulting banking system.
    - Credit-and-funding shock: banks recover 65 percent of lost funding (ρ = 0.35 replacement fraction), and a 50 percent discount rate applied to assets sold to cover funding gap (δ = 0.50).
  - Key contagion findings:
    - French banks are less inwardly vulnerable to cross-border interbank contagion since the last FSAP and somewhat less outwardly contagious.
    - Vulnerability from cross-border interbank loans has declined; contagion risk emanating from the French banking system remains among the highest in the world and has declined very slightly since the last FSAP.
    - Spillovers from France have greatest impact within the Euro Area (EA); spillovers to France emanate principally from outside the EA (United Kingdom, United States, Japan). Within the EA, the Netherlands, Ireland, Belgium, and Italy are susceptible to shocks from France; Germany is the primary inward source for France.
  - Note: domestic and cross-border holdings of banks in marketable securities comprise just 10.5 percent of banks assets.

### Implications for macroprudential policy and monitoring priorities
- Macroprudential signals and prescriptions:
  - When analysis reveals risk and capital gaps, capital should be increased in the respective sector by earnings retention or issuance of equity.
  - Any policy measure that closes capital and risk gaps (considering general equilibrium effects) could be suitable, even if certain macroprudential tools are not explicitly part of the model.
  - The model can be used as a signaling device:
    - Ex-ante and contemporaneously, to indicate when macroprudential intervention is needed.
    - Ex-post, to assess whether macroprudential action was successful in closing gaps.
- Historical policy-relevant episodes where more capital or intervention was needed:
  - Technology Crisis (firm capital), Global Financial Crisis (firm and bank capital), Sovereign Debt Crisis (mainly firm but also bank capital), and now (firm capital).
- Interconnectedness monitoring priorities and data recommendations:
  - Track evolution of common exposures over time, especially relationships strengthened since 2015.
  - Monitor bond funds’ overlap with other sectors and mixed funds’ central role in liabilities-side funding overlap.
  - Authorities should close remaining data gaps and monitor common exposure at different aggregation levels.
  - Remaining data gaps include absence of assets other than marketable securities and limited exposures between French-domiciled banks and insurance companies in available datasets; authorities planned completion of these data in 2019.

### Technical appendices: network and contagion methodology (highlights)
- Graph/network terminology: vertices, edges, directed/undirected graphs, degree, density, path, average distance.
- Direct network:
  - Edges w_ijt represent quantities lent from j to i at time t.
  - Direct in-strength: S_it^in = ∑ w_ijt_j.
- Common exposure network:
  - Define In_it and Out_it portfolio vectors.
  - In-flow common exposure edge: d_ijt^In = 2 − √{2(1 − C_In_it,In_jt)}.
  - Out-flow common exposure edge: d_ijt^Out = 2 − √{2(1 − C_Out_it,Out_jt)}.
  - Summary measures: in-strength K_it^In = ∑ d_ijt^In_j; out-strength K_it^Out = ∑ d_ijt^Out_i.
- Contagion framework (Espinosa-Vega and Solé (2011)):
  - Stylized balance sheet identity: ∑ x_ji + a_i = k_i + b_i + d_i + ∑ x_ij.
  - Credit shock: losses l_hi = λ x_hi (λ = loss-given-default).
  - Credit-and-funding shock: losses l_hi = δ ρ x_hi (with funding replacement fraction 1 − ρ and fire-sale distress δ).
  - Contagion index: CI_h = 100 ∑_i l_hi / ∑_i k_i.
  - Vulnerability index: VI_i = 100 ∑_{h≠i} l_hi / ((n − 1) k_i).
  - Simulation algorithm triggers sequential defaults until no more failures occur.

*Source: IMF staff chapter “1. Literature Review” and associated FSAP analysis (content unit 1fraea2019007).*

### 1. Literature Review _______________________________________________________________________________ 7

### 1. Literature Review

### Executive summary
- Macroprudential policy challenge:
  - Identifying growth of financial and macroeconomic variables above and below potential; gaps are unobserved (1).
- Model presented:
  - Captures (i) sectoral dynamics of firms and banks and feedbacks between them, (ii) capital and default risk dynamics of each sector, (iii) capital and risk gaps (deviations from the welfare maximizing optimum), and (iv) provides a quantitative method for measurement (2).
- Definition of potential:
  - Potential levels of capital and default risk are those a central planner would choose where corporate and bank capital levels are flexible (3).
- Use for policy:
  - Framework is silent on implementation instruments but can be used as a signaling device to indicate when macroprudential action is necessary; any policy that closes capital and risk gaps considering general equilibrium effects can be evaluated with the model (4).
- Data and estimation:
  - Model is taken from French aggregate publicly available data and estimated with Bayesian estimation techniques (5).
- Key findings on default risk and capital:
  - Default risk fluctuates between being too high and too low; risk is too high in four episodes: prior to the Technology Crisis, prior to the Global Financial Crisis, prior to the Sovereign Debt Crisis, and now (6).
  - Corporate default risk gap was particularly elevated during the Technology Crisis and now, although default risk is now relatively low in absolute terms (6).
  - Banking sector default risk was particularly elevated during the Global Financial Crisis and appears now adequate (6).
  - Corporate capital gaps: four episodes where more corporate capital is needed marked by a positive gap: prior to the Technology Crisis, prior to the Global Financial Crisis, prior to the Sovereign Debt Crisis, and now (7).
  - Bank capital was too low during the Global Financial Crisis and the Sovereign Debt Crisis, has strengthened significantly, and appears now to be adequate (7).
- Policy implication:
  - Firms should be encouraged to strengthen their equity capital base by retaining earnings or issuing equity; this could be done also indirectly by publishing related research (8).

*Prepared by Fabian Lipinsky and Mirela Sorina Miescu (chapter note).*

### Motivation and contribution
- Motivation:
  - Macroprudential policy needs measures of deviations from potential for financial variables (capital, default risk of borrowers and lenders); these gaps are unobserved but crucial for policymakers (Motivation bullets).
  - Global financial crisis highlighted importance of monitoring balance sheets and role of risk and uncertainty shocks in business cycles.
  - If capital is too low and default risk too high, policy responses could increase capital base or decrease debt.
- Contribution:
  - Presents a macrofinancial structural model capturing:
    - Sectoral dynamics of nonfinancial firms and financial intermediaries and their feedbacks (9).
    - Default risk in each sector and measurement via corporate and financial credit spreads (9).
    - Capital and risk gaps and a quantitative method to measure them in a theoretically sound framework (9).

### Literature and theoretical foundation
- Core references and framework lineage:
  - Based on Christiano, Motto and Rostagno (2014) focusing on balance sheet dynamics of firms and firm default; highlights idiosyncratic risk as an important business cycle driver (10).
  - Extended to include defaulting financial intermediaries and financial intermediary debt investors; in this framework aggregate risk is an important driver of financial intermediary PD (11).
  - Follows the financial accelerator literature where capital is accumulated mostly from retained earnings, implying sticky capital; welfare-maximizing capital and risk measures are calculated as benchmarks (12).

### High-level model summary
- Purpose:
  - Assess how capital and default risk of nonfinancial firms and financial intermediaries change over time and whether capital and risks are too low or too high, providing signals for macroprudential policy (13).
- Model structure (13):
  - Core: a standard real business cycle model enhanced with a financial system of three agent sets: financial intermediaries, financial investors, and nonfinancial firms.
  - Agents: Financial intermediaries borrow from financial investors and on-lend to nonfinancial firms; financial investors hold equity of firms and intermediaries.
  - Risks:
    - Nonfinancial firms face idiosyncratic and aggregate productivity shocks.
    - Firm PD depends on balance sheet strength, aggregate productivity, and the CDF of idiosyncratic productivity.
    - Financial intermediary PD depends on balance sheet strength of borrowers and intermediaries and the distribution function of aggregate productivity.
  - Capital gaps:
    - Firms and intermediaries accumulate capital from retained earnings via an exogenously given law of motion.
    - Optimal (welfare-maximizing) capitalization trades off tax benefits of debt and cost of default.
    - Capital gap = welfare-maximizing optimal capital − own capital accumulated out of retained earnings (13).
  - Deviations from potential:
    - Solution with sticky capital (retained earnings) is compared to solution with optimal capital to determine deviations (capital, default risk, lending, funding).
  - Empirical application:
    - Model is fitted to French macroeconomic and financial time series; parameters estimated to maximize the likelihood of observing the data; model also solved for the welfare-maximizing optimum and deviations are calculated (13).

### Detailed description — core and financial system
- Core of the model:
  - Standard RBC model with exogenous government spending and investment adjustment cost; four macro shocks: total factor productivity, government spending, investment efficiency, and labor market tightening (disutility of labor) (15).
- Financial system agents (16–17):
  - Three agent sets: nonfinancial firms, financial intermediaries, financial investors (households).
  - Financial intermediaries use own funds and funds from financial investors (deposits and wholesale funding) to issue loans; nonfinancial firms use loans plus own funds to acquire assets (17).
- Funding constraints and shocks (18):
  - Acquisition of assets by nonfinancial firms gives rise to two financial shocks:
    - Cross-sectional return-on-asset variance (idiosyncratic risk).
    - Aggregate return-on-asset variance (aggregate risk) (18).

### Nonfinancial firms: cashflows, default, PD
- Firm cashflows and default mechanics (19–21):
  - Firms have initial outlay of own funds; next period they receive a return on invested assets and pay back debt; returns are subject to idiosyncratic and aggregate productivity shocks; income taxed; future cashflows discounted at a stochastic discount factor (19).
  - Firms default if end-of-period assets < liabilities, equivalently if idiosyncratic productivity falls below a default threshold (20).
  - Firm PD depends on firms’ balance sheet strength, aggregate productivity, and the cumulative distribution function of idiosyncratic productivity (21).

### Financial intermediaries: loan portfolio, loss measurement, PD
- Intermediary cashflows and loan returns (22–24):
  - Intermediaries have initial outlay of own funds; next period they receive loan portfolio returns and pay back funding; income taxed (22).
  - Loan portfolio return components:
    - Interest and principal from non-defaulting firms and recovered assets from defaulted firms, net of screening costs (23).
  - Expressions rewritten yield model-based loss-given default (LGD), expected losses, and model-based loss rate (24).
- Intermediary default and PD (25):
  - Intermediaries default if end-of-period assets < liabilities, equivalently if aggregate productivity falls below a default threshold.
  - PD of financial intermediaries depends on borrower and bank balance sheet strength and the distribution of aggregate productivity; simplification assumes aggregate productivity takes a fixed value above and below the default threshold (25).

### Financial investors
- Role (26):
  - Financial investors (mutual funds) hold equity in nonfinancial firms and financial intermediaries and provide funding (deposits, wholesale funding) to intermediaries.
  - Investors receive funding back if intermediary doesn’t default and receive assets of intermediaries in case of default.
  - Expected cashflows of financial investors described in model (26).

### Capital accumulation and optimization
- Capital dynamics (27–29):
  - Firms and intermediaries accumulate capital out of retained earnings per an exogenously given law of motion; a constant fraction of earnings is distributed as dividends, remainder retained (27).
  - Firms maximize cashflow taking intermediary and investor cashflows as given, with zero-profit assumptions for intermediaries and investors (28).
  - Given initial capital positions, firms scale up balance sheets financed through liabilities to maximize future profits (29).

*Source: IMF staff chapter “1. Literature Review” (content unit 1fraea2019007).*

### 30.      The optimal lending rate and financial intermediaries’ cost-of  -funding is determined

### 1fraea2019007 - 30.      The optimal lending rate and financial intermediaries’ cost-of  -funding is determined

### Financial frictions and optimality
- Three financial frictions in the model:
  - Income of nonfinancial firms and financial intermediaries is tax-deductible, while income of financial investors is not. More intermediated funding between financial intermediaries and financial investors results in a tax-benefit ("the benefit of financial intermediation").
  - Defaults of nonfinancial corporates trigger a screening cost.
    - The optimal lending rate is determined by trading off the benefit of financial intermediation associated to non-defaulting loans versus the cost of default.
  - Defaults of financial intermediaries result in lower productivity.
    - Financial intermediaries’ cost-of-funding is determined by trading off the benefit of financial intermediation associated with more funding versus the productivity loss in case of default.
- Nonfinancial firms maximize with respect to three choice variables: financial intermediaries’ funding (hence nonfinancial firms’ liabilities), lending rates, and financial intermediaries’ cost-of-funding.
- Equity capital adjustment cost could be added as an alternative to exogenous capital accumulation (capital is sticky in both approaches). Identified capital gaps may differ if adjustment cost were used.

### The capital gaps (concept and calculation)
- Capital is sticky and accumulated out of retained earnings (as in Christiano, Motto and Rostagno (2014)).
- Comparison between model solution (sticky capital) and welfare-maximizing optimal solution (flexible capital) yields capital gaps.
  - The optimal welfare-maximizing problem solves for five choice variables (including optimal levels of capital for nonfinancial firms and financial intermediaries), versus three in the model solution.
- Capital gaps are defined as the difference between welfare-maximizing optimal capital and own capital accumulated out of retained earnings — analogous to an output gap between sticky and flexible prices.
- Deviations of capital from potential imply deviations of risk from potential (nonfinancial firm default risk and financial intermediary default risk).
- Deviations from potential of capital, default risk, lending, and funding provide signals for policymakers when to tighten or loosen macroprudential policies.
- Default thresholds and constraints are embedded in the welfare maximization subject to funding constraints, FI participation and capital constraints, and wholesale FI debt investor participation constraints (mathematical expressions preserved in source).

### Empirical strategy and application to France
- Data fitted: quarterly macroeconomic and financial time series between 2000:Q4 and 2018:Q2.
  - Macroeconomic time series: output, investment, employment, and consumption (all demeaned percentage changes).
  - Financial time series: corporate credit spread (demeaned percentages), financial credit spread (demeaned percentages), and aggregate bank capital (demeaned percentage changes).
- Estimation approach: Metropolis-Hastings algorithm (Bayesian estimation) to find parameters that maximize likelihood of observed data. Standard parameters were calibrated.
- Models with the frictions presented improve model fit relative to the standard RBC model.
- Shock decomposition identifies historical shock realizations that generated the data:
  - Macroeconomic shocks: total factor productivity (‘ey’), government spending (‘eg’), investment efficiency (‘ei’), and labor market tightening (disutility of labor, ’en’).
  - Financial shocks: cross-sectional return-on-asset variance of nonfinancial firms (idiosyncratic risk, ‘eF’), aggregate return-on-asset variance of nonfinancial firms (aggregate risk, ‘eB’), and financial intermediary income shock (‘ek’), altering capital accumulation of financial intermediaries out of retained earnings.
- The model is solved also for the optimum (the potential), and deviations from potential are reported for:
  - Capital of nonfinancial firms
  - Capital of financial intermediaries
  - Default risk of nonfinancial firms
  - Default risk of financial intermediaries
- Gaps are proposed as guidance for macroprudential policy (signaling when intervention is needed ex-ante, contemporaneously, and for ex-post assessment of success in closing gaps).

### Results — gap analysis: default risk dynamics and drivers
- Empirical findings:
  - Figures 2 and 3 (as referenced) show demeaned probability of default of the firm and banking sector (black line).
  - Default risk spiked during the 2009 Global Financial Crisis and during the 2012 European Sovereign Debt Crisis.
  - According to the model, default risk is currently low, as credit spreads are low.
- Shock decomposition: main drivers of default probabilities:
  - Corporate default risk is driven significantly by cross-sectional, idiosyncratic risk (‘eF’ — yellow bars).
  - Banking default risk is mostly driven by aggregate risk (‘eB’ — light blue bars).
  - Productivity shocks (‘ey’ — blue bars) are also important contributors.
- Figure notes (numeric precisions preserved):
  - For corporate PD figure: the y-axis denotes percentages (0.01 equals to 1 percent).
  - For banking PD figure: the y-axis denotes percentages (0.01 equals to 1 percent).

### Results — risk gaps
- Deviations of probability of default from the welfare-maximizing optimum (risk gaps) fluctuate over time between being too high and too low.
- Risk is too high (risk gap positive) during four episodes:
  - prior to the Technology Crisis,
  - prior to the Global Financial Crisis,
  - prior to the Sovereign Debt Crisis,
  - and now.
- Specific observations:
  - Default risk in the corporate sector was particularly elevated during the 2002 Technology Crisis, and now (although now low in absolute terms).
  - Banking sector default risk was particularly elevated during the Global Financial Crisis and appears now adequate.
- Figure notes (numeric precisions preserved):
  - Corporate risk gap figure: the y-axis denotes percentages (0.005 equals to 0.5 percent excessive default risk in absolute terms).
  - Banking risk gap figure: the y-axis denotes percentages (0.005 equals to 0.5 percent excessive default risk in absolute terms).

### Results — capital gaps
- Deviations of firm capital and banking capital from the welfare-maximizing optimum shown in percentage deviations.
- Episodes where more corporate capital is needed (positive gaps): prior to the Technology Crisis, prior to the Global Financial Crisis, prior to the Sovereign Debt Crisis, and now.
- Bank capital observations:
  - Bank capital was too low during the Global Financial Crisis and the Sovereign Debt Crisis.
  - Bank capital appears now to be adequate and has improved significantly.
- Figure notes (numeric precisions preserved):
  - Corporate capital gap figure: the y-axis denotes percentages (0.05 equals to a 5 percent gap relative to the total market cap).
  - Banking capital gap figure: the y-axis denotes percentages (0.2 equals to a 20 percent gap relative to the nominal amount of aggregate capital).

### Results — macro stress test (scenario and outcomes)
- Scenario generation:
  - The worst historical downturn is the Global Financial Crisis. The stress test applies structural shocks from 2008–2011 to the current economy to simulate 2018–2021.
  - Shocks of the Global Financial Crisis during 2008–2011 were applied now to simulate the macro stress test between 2018–2021.
- Stress test results:
  - Bank capital is about adequate under the stress scenario; more corporate capital is needed under adverse conditions.
  - Firms are more vulnerable than banks at the current juncture.
  - Under adverse conditions, corporate and banking risk gaps are elevated:
    - The corporate risk gap reaches unprecedented highs under the stress scenario.
    - The bank risk gap is considerably lower than during the Global Financial Crisis, even if stress intensifies.
  - Figure 8 (referenced) shows gap analysis including a dotted red line simulating a crisis period of magnitude like the Global Financial Crisis.
  - Figure notes: for risk, 0.005 equals to 0.5 percent excessive default risk in absolute terms; for capital, 0.2 equals to a 20 percent gap relative to the nominal amount of aggregate capital. Grey bars highlight recession periods.

### Implications for macroprudential policy setting
- When analysis reveals risk and capital gaps, capital should be increased in the respective sector by earnings retention or issuance of equity.
- Any policy measure that closes capital and risk gaps (considering general equilibrium effects) could be suitable, even if certain macroprudential tools are not explicitly part of the model.
- The model can be used as a signaling device:
  - Ex-ante and contemporaneously, to indicate when macroprudential intervention is needed.
  - Ex-post, to assess whether macroprudential action was successful in closing gaps.
- Historical policy-relevant episodes where more capital or macroprudential intervention was needed:
  - the Technology Crisis (firm capital),
  - the Global Financial Crisis (firm and bank capital),
  - the Sovereign Debt Crisis (mainly firm but also bank capital),
  - and now (firm capital).
- Note: A more forward-looking result may be obtained by using leverage or coverage ratios instead of credit spread data.

### Interconnectedness and contagion — executive summary (selected points)
- The French financial system features globally active financial conglomerates with business lines across banking, insurance, and asset management; cross-sector and cross-border links are policy-relevant.
- Three datasets and three approaches were used to assess interconnectedness:
  - direct exposure via network visualization and summary statistics;
  - clusters of common exposure via portfolio overlap analysis;
  - contagion exercise using interbank loan data for banks to compensate for lower coverage in marketable securities dataset.
- Stylized findings on exposures and portfolio shifts:
  - Across financial entities, exposures to non-financial corporations (NFCs) have increased.
  - Banks reduced exposures to domestic monetary financial institutions (MFIs) in favor of other domestic nonbank financials and French government securities.
  - Insurers reduced exposures to foreign non-MFIs and increased holdings of securities issued by the French government and foreign nonbank financials.
  - Investment funds shifted away from domestic MFIs and the French government but increased domestic and cross-border holdings across other counterparty sectors, notably non-MFIs.
- Network and common exposure findings:
  - Increased individual diversification but cross-border exposure shifts have produced stronger clusters of common exposure.
  - Two main overlapping-portfolio clusters:
    - cluster centered on mixed funds (includes equity, bond, and other funds);
    - cluster consisting of banks and insurers.
  - Clusters are linked through common exposures of insurers and mixed funds.
  - Common exposure among fund sectors has been increasing, potentially increasing aggregate risk.
- Contagion and interbank findings:
  - French banks are less inwardly vulnerable to cross-border interbank contagion since the last FSAP, and somewhat less outwardly contagious.
  - Vulnerability from cross-border interbank loans has declined; spillovers from France have greatest impact within the EA, while spillovers to France emanate principally from outside the EA.
- Data and monitoring recommendations:
  - Authorities should close remaining data gaps and monitor common exposure at different aggregation levels.
  - Gaps include absence of assets other than marketable securities and limited exposures between French-domiciled banks and insurance companies in available datasets.
  - Reliable financial group identification and compilation of regulatory capital data at the entity level would permit more detailed analysis of domestic exposures.
  - Authorities’ planned completion of this data during 2019 is important.

*Source: IMF staff calculations and model results as presented in the chapter.*

### 57.      The FSAP analysis exploits three different datasets to obtain complementary

### 1fraea2019007 - 57.      The FSAP analysis exploits three different datasets to obtain complementary

### Data overview and datasets
- Three datasets exploited to obtain complementary perspectives on interconnectedness:
  - Unilateral (domestic and cross-border) exposures in marketable securities for banks, insurers, and funds (almost complete for insurers and funds; banks hold a relatively small share of assets in marketable securities).
  - Bilateral reporting dataset for domestic assets and liabilities between French financial entities (lower coverage but more granular, permitting assessment of network properties).
  - BIS consolidated banking statistics to examine interbank loans at the country aggregation level for banks.
- Marketable securities comprise equity, debt securities, and fund shares; assets are assumed to be marked-to-market.
- Data coverage and timing:
  - Unilateral dataset coverage relative to total assets: banks 10.5 percent, insurers 92.7 percent, funds 96.3 percent.
  - Bilateral dataset coverage relative to total assets: banks 2.2 percent, insurers 29.8 percent, funds 25.7 percent.
  - Unilateral dataset covers EUR 4.8 trillion in assets (of which EUR 1.1 trillion to French financial entities).
  - Bilateral dataset covers EUR 835 billion in assets between French financial entities.
  - Data frequency and periods: banks annually 2008–2017; insurers annually 2011–2017 (structural break with Solvency II in January 2016); funds quarterly 2008–2017; bilateral annual 2010–2016.
- Data gaps and limitations:
  - Exposures between banks and insurance companies in the bilateral dataset are missing.
  - Exposures in instruments other than marketable securities are missing in both unilateral and bilateral datasets.
  - Domestic interbank exposures across all instruments are missing from the marketable-securities bilateral dataset: marketable securities of other French banks constitute 1.6 percent of total assets, while exposure through interbank loans is much higher at 31 percent of total assets.
  - Insurers hold 11 percent of their balance sheet in securities issued by French banks; bilateral bank–insurer securities exposures are therefore material but incompletely reported.
  - Data on large interbank exposures are restricted by the ECB and were not available for this analysis.
  - Work in progress: correspondence between SIREN and LEI to report bilateral exposures between banks and insurance companies; HCSF aims to complete cross-sector interconnectedness dataset within 2019.

### Cross-border interconnectedness — aggregate direct exposures and trends
- Aggregate magnitude and evolution:
  - Total exposures in marketable securities across banks, insurance companies, and funds: EUR 4.8 trillion, an increase of 27 percent since 2011.
  - By sector in 2017 (change since 2011):
    - Insurance companies: EUR 2.4 trillion (+36 percent since 2011).
    - Banks: EUR 896 billion (-12 percent since 2011).
    - Funds: EUR 1.5 trillion (+30 percent since 2011).
- Cross-border exposures in 2017 and changes since 2011:
  - Insurers cross-border exposures: EUR 1.15 trillion (+25 percent since 2011); cross-border share of insurers’ total exposures 49 percent (-2 percent since 2011).
  - Funds cross-border exposures: EUR 784 billion (+47 percent since 2011); cross-border share of funds’ total exposures 52 percent (+8 percent since 2011).
  - Banks cross-border marketable securities holdings reduced to EUR 375 million in 2017 (-39 percent since 2011); cross-border exposures remain a steady 42 percent share of banks’ marketable securities holdings since 2011.
- Geographic footprint:
  - French banks, insurers, and funds maintain diversified global exposures concentrated in the Euro Area, the United States, and the United Kingdom.
  - The highest individual country exposure is to Luxembourg: EUR 337 billion in marketable securities holdings.
- Composition by counterparty sector and instrument:
  - Shift toward domestic and foreign NFCs across banks, insurers, and funds.
  - Banks’ most substantial individual exposure: domestic exposures to the French government; increasing exposure to domestic nonbank financials.
  - Insurers: significant increases in exposures to the French government and to nonbank financials; insurers’ cross-border exposures concentrated in MFIs and NFCs.
  - Funds: declining exposures to domestic MFIs since 2011 offset by increased exposure to NFCs and other French funds; fastest-growing fund exposures are to foreign non-MFIs and NFCs.
  - Instrument composition:
    - Banks: debt securities comprise 80 percent of domestic positions and 69 percent of cross-border positions.
    - Insurance companies: debt securities comprise 76 percent of domestic and 72 percent of cross-border securities; domestic exposures in equity have increased since 2015 (note structural break for insurers with Solvency II in 2016).
    - Funds (domestic positions since 2012): debt 33 percent, equity 33 percent, fund shares 34 percent.
    - Funds (abroad): debt 55 percent, equity 33 percent, foreign fund shares 12 percent.

### Network properties and common exposures
- Direct exposure network patterns:
  - Number of counterparties (vertices) for banks has remained relatively constant while number of connections (edges) has declined, indicating fewer banks invested in cross-border counterparties and a retreat from cross-border exposures.
  - Insurance companies and funds show increased numbers of counterparties and connections, suggesting increased individual diversification in cross-border exposures (interpretation for insurers should consider Solvency II structural break).
- Common exposure and clustering:
  - Increased individual diversification in cross-border exposures coexists with increasingly strong clusters between equity, bond, mixed, and other funds, and between banks, insurers, and mixed funds — implying potential common-exposure risks despite diversification of direct exposures.
  - Domestic out-flow common exposures are organized around bond and mixed funds:
    - First cluster: bond, other, and money market funds.
    - Second cluster: mixed funds, banks, and insurers.
  - Domestic in-flow common exposure consists of one cluster comprised of all funds (likely influenced by the structure of the data).
- Contagion and interbank spillovers:
  - French banks are less inwardly vulnerable to cross-border interbank contagion since the last FSAP and are also less outwardly contagious.
  - Spillovers from France have the greatest impact within the Euro Area (EA), while spillovers to France emanate principally from outside the EA.
  - The analysis complements marketable-securities-based interconnectedness with an interbank contagion exercise using BIS consolidated statistics to capture loan-based linkages omitted from securities data.

### Findings, methodological notes, and planned extensions
- Findings summary:
  - Shifting composition of total exposures domestically and across borders: increased exposures to NFCs across banks, insurers, and funds.
  - Banks have invested away from domestic MFIs; insurers have increased exposure to the French government; funds have reduced exposure to domestic MFIs and the French government while increasing exposures in other sectors domestically and abroad.
  - Domestic network organized in hubs: insurers and banks form hubs; funds form hubs around insurers through liabilities and around banks through assets.
- Methodological notes:
  - Common exposure assessed using the Giudici, Sarlin and Spelta (2017) framework.
  - Contagion exercise follows Espinosa-Vega and Solé (2011) on an interbank loan network where adequate data exist.
  - Comparative interpretation of insurer time series should consider the structural break from implementation of Solvency II in January 2016.
- Planned data and analytical extensions by authorities:
  - HCSF plans to expand the bilateral dataset to incorporate direct exposures in marketable securities between banks and insurance companies.
  - HCSF is incorporating direct and indirect exposures in an initial model to assess shock propagation through two main channels: losses through mark-to-market accounting of securities; and losses due to potential defaults. Behavioral aspects (asset fire sales and investor runs) will eventually be incorporated. The authorities aim to complete this work in 2019.
  - Work remains to identify entity-level group membership (which banks, insurers, and funds belong to specific financial groups) to examine within- and across-group trends.

*Source: Excerpts from the FSAP chapter on interconnectedness (unilateral and bilateral datasets, HCSF), as provided in the supplied content.*

### 73.      An assessment of risks stemming from common exposures in assets requires a more

### An assessment of risks stemming from common exposures in assets requires a more careful examination of direct exposures

### Cross-border common exposures: methodology and significance
- Methodology: uses the linear transformation of distance to define similarity between portfolio vectors (out-flow common exposure for assets, in-flow for liabilities) following Giudici, Sarlin and Spelta (2017).
- Rationale: distance is used because the objective is to identify clusters that are most similar; higher out-flow common exposure implies higher correlation and lower distance.
- Significance testing: t-statistic 푡 = 푟 √(푛−2) / √(1−푟2). For 2026 degrees of freedom, the one-sided test critical values are 2.33, 1.96, and 1.28 corresponding to out-flow common exposures (correlations) of 0.64 (0.07), 0.62 (0.05), and 0.61 (0.04) at the 1 percent, 5 percent, and 10 percent levels respectively.
- For out-flow cross-border assets, significance is achieved at all conventional levels for every out-flow common exposure measure reported.

### Cross-border common exposure: key findings and statistics
- Funds sector shows relatively high common exposure internally and forms a cluster:
  - Equity–Mixed: 1.32
  - Bond–Mixed: 1.24
  - Mixed–Other: 1.40
  - Equity–Other: 1.17
- Insurance–bank and insurer–mixed fund common exposures:
  - Insurer–Bank: 1.20
  - Insurer–Mixed: 1.19
- Bond funds have highly significant overlap with every other sector.
- Interpretation: common exposure measures the proportional overlap of portfolios; shocks affecting one entity are likely to affect others with high common exposure.
- Trend: common exposures for significant pairs have been increasing since 2015; a cluster is forming among equity, bond, mixed, and other funds. Insurers are on the periphery of this cluster via mixed funds; banks connect via insurers. Since 2015, the strength of these relationships has been increasing, implying shocks to cited fund sectors will also impact insurers; banks are more likely affected by shocks common to insurers.

### Domestic interconnectedness — dataset coverage and direct exposures
- Total domestic exposures in marketable securities held by domestic financial entities: EUR 1.1 trillion.
- Bilateral reporting dataset coverage: EUR 835 billion of assets.
- Bilateral dataset coverage shares of total assets by sector in sample:
  - Banks: 2.2 percent of banks’ total assets
  - Insurers: 29.8 percent of insurers’ assets
  - Funds: 25.7 percent of funds’ assets
- Network size (2017, bilateral dataset):
  - Funds: 9,670
  - Banks: 132
  - Insurance companies: 130
  - Total vertices: 9,932
  - Total edges (exposures from a bank or insurance company to a fund): 61,348
- Network metrics:
  - Density: 0.06 percent
  - Average distance: 3.24 (varies between 1 and 14)
  - Degree distribution: 75 percent of entities connected to at most 10 others in 2017; a very small number of entities have more than 1000 links (three banks and six insurance companies).
- Interpretation: declining number of vertices and edges over the sample suggests consolidation and increased interconnectedness; low density with low average distance implies hub-like structure (a few well-connected banks and insurers, many sparsely connected funds).

### Domestic common exposures: findings and clusters
- Statistical threshold: a common exposure of 1.07 is significant at the 1 percent level (in the bilateral dataset context).
- Out-flow (asset) common exposures:
  - On average, diversification of idiosyncratic risk does not lead to increased aggregate risk through common exposure; average out-flow common exposure reaches low levels of significance for a subset of entities.
  - High common exposure between:
    - Bond, Other, and Money Market funds (examples: Bond–Other or Bond–MMF = 1.40 in Table 2)
    - Banks and Insurers domestic common exposure: 1.30
    - Banks–Mixed: 1.20
    - Insurers–Mixed: 1.52
  - Two domestic clusters identified:
    - Cluster 1: bond funds with other funds and MMFs (1.40)
    - Cluster 2: banks and insurers with significant overlap and links to mixed funds
- In-flow (liabilities) common exposures:
  - Funds display significant funding overlap, especially centered around mixed funds:
    - Mixed–Equity: 1.77
    - Mixed–Bond: 1.80
    - Mixed–Hedge: 1.82
    - Mixed–Other: 1.76
    - Mixed–MMF: 1.40 (MMFs on the periphery but still significant)
  - Banks and insurance companies show no significant funding overlap with the rest of the financial system in the bilateral dataset.
  - The liabilities-side results are likely influenced by the dataset structure where funds are connected through a few insurers on the liabilities side.

### Implications and monitoring priorities
- Funds industry risk: although individual fund sectors may show increased direct cross-border exposures and apparent idiosyncratic diversification, aggregate industry risk is elevated due to growing portfolio composition overlap (common exposure) across fund sectors.
- Cross-sector contagion pathways:
  - Shocks to fund clusters (equity, bond, mixed, other) are likely to transmit to insurers via common cross-border exposures and to banks indirectly via insurers.
  - Domestic portfolios show overlapping exposures among fund subtypes and strong bank–insurer overlap, indicating potential amplification channels for domestic shocks.
- Monitoring recommendations:
  - Track evolution of common exposures over time, with particular attention to relationships that have strengthened since 2015.
  - Monitor bond funds’ overlap with other sectors given their extensive common exposure.
  - Monitor mixed funds’ role as a central node in liabilities-side funding overlap among fund types.

*Source: Haut Conseil de Stabilité Financière (HCSF) and IMF staff.*

### 88.      Some recent trends of out- and in-flow common exposure should be monitored. Figure

### 1fraea2019007 - 88.      Some recent trends of out- and in-flow common exposure should be monitored. Figure

### Recent trends in out- and in-flow common exposure
- Since 2013, both clusters previously identified by significant out-flow common exposure have shown an upward tendency, indicating portfolios are increasingly overlapping.
- The cluster identified by significant in-flow common exposure has been increasing in proximity of funding composition over the entire sample.
- Two interlinked clusters of elevated common exposure risk were identified:
  - Cluster 1: centered around mixed funds and includes equity, bond, and other funds.
  - Cluster 2: consists of banks and insurers.
  - The clusters are linked through the common exposure of insurers and mixed funds.
- Common exposure among the cited fund sectors has been increasing steadily, suggesting that diversification of idiosyncratic counterparty risk may be generating additional aggregate risk.
- Trends can be monitored by examining more narrowly how portfolios comove (Figure 16 shows index trends for Common Exposure Trends in Domestic Assets and Domestic Liabilities for 2010–2016).

### Data caveats and interpretation of domestic common exposures
- Results on domestic common exposures should be interpreted cautiously:
  - The data do not cover significant balance sheet exposures for banks, so the impact of high common exposure may not be very consequential.
  - The disaggregated nature of the data means some results may be driven by group ownership structure; information on groups was not made available to the mission.
- Unit-linked insurance is not distinguishable in the domestic dataset; some risk attributed to insurance companies will be borne by end-investors. Thus, out-flow common exposure of insurers may be slightly overstated, particularly with respect to mixed funds.
- Empirical notes:
  - For 14 degrees of freedom, the one-sided test has 1 percent, 5 percent, and 10 percent critical values at 2.62, 1.76, and 1.35; these correspond to out- and in-flow common exposures (correlations) of 1.07 (0.57), 0.93 (0.43), and 0.85 (0.34).
  - Hauton and Héam (2015) find diversification occurs at the group level, so common exposure risks may be slightly overstated. Benhami et al. (2018) report intra-group exposures do not exceed 4 percent of total group assets.

### Structure of domestic direct exposures and hub formation
- Domestic direct exposures imply formation of hubs around insurers and banks, with consequences for common exposures:
  - Funds form hubs around insurers through liabilities and around banks through assets.
  - On the asset side, domestic common exposures are organized around bond and mixed funds:
    - First asset cluster: bond, other, and money market funds.
    - Second asset cluster: mixed funds, banks, and insurers.
  - On the liabilities side, all funds are clustered around mixed funds with significant common exposure (partially attributable to dataset structure).
- These clusters generally display an upward tendency since 2013 and should be monitored.

### Cross-border interbank contagion: data, assumptions, and main results
- Data and approach:
  - Interbank loan exposures (BIS consolidated banking statistics) were paired with regulatory capital data to assess cross-border interbank contagion at the international level.
  - Interbank loan exposures and capital data were used for 2012:Q4 and 2018:Q4 in contagion analysis.
  - The methodology follows Espinosa-Vega and Solé (2011); sequential defaults among national banking sectors are triggered if capital is insufficient to cover losses and the algorithm stops once no more failures occur.
  - Countries included in the contagion analysis: Austria, Australia, Belgium, Canada, Switzerland, Chile, Germany, Spain, Finland, France, Greece, the United Kingdom, Hong Kong Special Administrative Region, Ireland, India, Italy, Japan, Korea, the Netherlands, Norway, Portugal, Sweden, Singapore, Turkey, and the United States.
- Modeling assumptions for shocks:
  - Credit shock: the banking system of a country defaults on its interbank loans with a loss-given-default of 100 percent.
  - Credit-and-funding shock: banks are only able to recover 65 percent of the lost funding from a default entity, and a 50 percent discount rate is applied on assets that a bank may be forced to sell to cover the funding gap.
- Key quantitative findings:
  - French banks are less vulnerable and slightly less contagious than during the last FSAP.
  - Vulnerability risk from cross-border exposures in interbank loans has declined significantly.
  - Contagion risk emanating from the French banking system remains among the highest in the world and has declined very slightly since the last FSAP.
  - Domestic and cross-border holdings of banks in marketable securities comprise just 10.5 percent of banks assets.
- Spillover patterns:
  - Spillovers from France have the greatest impact within the Euro Area (EA); within the EA, the Netherlands, Ireland, Belgium, and Italy are susceptible to shocks from France; outside the EA, the United Kingdom is similarly susceptible.
  - Spillovers to France emanate principally from outside the EA; France is susceptible primarily to shocks from the United Kingdom, the United States, and Japan. Within the EA, Germany is the primary source of inward vulnerability for France.
  - Spillovers to France have declined since the last FSAP, notably with respect to the United States, translating into an overall lower vulnerability index in simulations.

### Concluding findings, trends, and data gaps
- Shifts in exposure composition:
  - Across the board, there have been large investments in marketable securities issued by domestic and foreign NFCs.
  - Banks have reduced exposures to domestic MFIs in favor of other domestic nonbank financials and the French government.
  - Insurers have significantly drawn down exposures to foreign non-MFIs in favor of securities issued by the French government and foreign nonbank financials.
  - Funds have invested away from domestic MFIs and the French government but increased domestic and cross-border holdings across every other counterparty sector, most notably non-MFIs.
- Monitoring priorities:
  - Monitor the two identified clusters (mixed funds + funds; banks + insurers) and the linkage through insurers and mixed funds because common exposure is trending upward since 2013.
  - Monitor fund-sector common exposure increases, as diversification at the idiosyncratic level may raise aggregate risk.
- Remaining gaps and authorities’ agenda:
  - Significant gaps remain: domestic bilateral holdings of marketable securities are not currently available for banks and insurers, though the French authorities are well underway in completing this dataset.
  - Gaps across other types of exposures could not be surmounted.
  - The French authorities are working to complete data gaps and progress on contagion research to monitor cross-sector and intra-group holdings more comprehensively.

*Source: Haut Conseil de Stabilité Financière (HCSF) and IMF staff; Bank for International Settlements (BIS) and IMF staff calculations.*

### Appendix III. Interconnectedness and Contagion Technical

### Appendix III. Interconnectedness and Contagion Technical

### A. Graphical Model Terminology
- A graph or network is a collection G of vertices V and edges E.
- Directed graph: edges are oriented between vertices. Undirected otherwise.
- Degree of a vertex: number of edges connected to the vertex.
  - For directed graphs, degree can refer alternatively to edges that originate or conclude at a vertex, or both.
- Fully connected graph: all vertices are connected to all other vertices.
- Density of a graph: ratio of the total number of edges to the total possible number of edges (i.e., those in a fully connected graph).
- Path between two vertices: set of edges connecting those vertices.
- Average distance: average number of edges along the shortest paths for all pairs of vertices in a graph.

### B. Direct and Common Exposure Networks
- Systemic risk: risk is systemic if it simultaneously affects many market participants so losses spread throughout the system.
- Network models: used to understand contagion effects; methodology implemented follows Giudici, Sarlin and Spelta (2017).

Direct Network
- Definition: graph comprised of edges w_ijt representing quantities lent from j to i at time t.
- Direct in-strength of entity i:
  - S_it^in = ∑ w_ijt_j
- Direct out-strength of entity j: symmetrically defined (notation in source for out-strength is shown but must be read as symmetric to in-strength).

Common Exposure Network
- Definition: graph where edges represent a function of the correlation between vertices.
- Define vectors:
  - In_it ∈ R^{1×N}: quantities lent to entity i by all other entities at time t.
  - Out_it ∈ R^{1×N}: quantities lent by entity i to all other entities at time t.
- In-flow common exposure network edges:
  - d_ijt^In = 2 − √{2(1 − C_In_it,In_jt)}
  - where C_In_it,In_jt is the correlation between funding compositions of i and j (funding similarity).
- Out-flow common exposure network edges:
  - d_ijt^Out = 2 − √{2(1 − C_Out_it,Out_jt)}
  - where C_Out_it,Out_jt is the correlation between the portfolio compositions of i and j (portfolio similarity).
- Summary measures for common exposure network:
  - In-strength: K_it^In = ∑ d_ijt^In_j
  - Out-strength: K_it^Out = ∑ d_ijt^Out_i

### C. Contagion Risk
- Contagion risk assessment: examine bilateral exposure data alongside regulatory capital data to analyze when direct and common exposures become costly.
- Methodology: follows Espinosa-Vega and Solé (2011) — stylized model of banks’ balance sheets commonly used across FSAPs.

Stylized balance sheet identity for bank i:
- ∑ x_ji + a_i = k_i + b_i + d_i + ∑ x_ij
  - where x_ji represents i claims on j
  - a_i represents other assets
  - k_i represents capital
  - b_i represents borrowing excluding interbank loans
  - d_i represents deposits

Credit shock (loss-given-default λ)
- If h defaults, effect on i:
  - ∑_{j≠h} x_ji + (1 − λ) x_hi + a_i = (k_i − λ x_hi) + b_i + d_i + ∑ x_ij
- If capital is insufficient to cover losses, i will default.

Credit-and-funding shock (funding replacement fraction 1 − ρ, fire-sale distress δ)
- i able to replace fraction 1 − ρ of lost funding from defaulted entity h.
- i must sell assets worth (1 + δ) ρ x_ih to cover funding shortfall.
- Effect on i:
  - ∑ x_ji − (1 + δ) ρ x_hi + a_i = (k_i − δ ρ x_hi) + b_i + d_i + ∑ x_ij − ρ x_ih
- i will default if it cannot absorb the funding shortfall cost induced by a fire sale of its assets.

Algorithm and indices
- Simulation: algorithm simulates failures of entities and stops when no more failures occur.
- Outcomes:
  - Contagion (outward spillover): assessed by number of failures suffered or induced.
  - Vulnerability (inward spillover): assessed similarly.
- Contagion index of entity i with initial default h:
  - CI_h = 100 ∑_i (l_hi) / ∑_i k_i , where losses l_hi of i from default of h can be:
    - λ x_hi under a credit shock
    - δ ρ x_hi under a credit-and-funding shock
- Vulnerability index of entity i:
  - VI_i = 100 ∑_{h≠i} l_hi / ((n − 1) k_i)
  - where n represents the number of entities in the network

*Source: Appendix III. Interconnectedness and Contagion Technical*

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_Source: https://www.imf.org/-/media/files/publications/cr/2019/1fraea2019007.pdf_