## _cr10244

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### Three main pillars of the stress testing analysis
- Balance-sheet based macroprudential stress tests (Section II)
  - Used publicly available financial statements and macroeconomic data to forecast financial firms’ capital needs.
  - Modeled how macroeconomic developments may affect the health of financial institutions and their lending capacity.
  - Did not account for default dependencies across institutions; may underestimate tail-risk (Figure 1).
- Macroprudential stress testing exercise with distress (Section III)
  - Accounted explicitly for distress dependencies across financial institutions using market-based CDS data.
  - Computed measures of probability of default and provided estimates of unexpected losses, interconnectedness and spillovers.
  - Noted limitation: CDS data may not fully capture true probability of default where few uninsured creditors bore distress.
- Estimates of government’s potential contingent liabilities implied by financial market prices (Section IV)
  - Used equity and CDS prices in a high-dimensional extension of contingent claims analysis (CCA).
  - Focused on expected shortfall: average density of extreme losses beyond the 95 percent Value-at-Risk (VaR) (Figure 1).

### Key findings and system-wide assessment
- Reliance and uncertainty:
  - Stress tests relied on publicly available information and models subject to considerable estimation uncertainty; lack of granular supervisory information was a constraint.
- Baseline outcome:
  - Under the baseline macro scenario, following substantive public and private capital injections, equity buffers appear satisfactory for the system as a whole.
- Vulnerabilities identified:
  - Some individual institutions, including a few of the smaller SCAP institutions, may be less well-positioned to absorb future losses through earnings.
  - Existing capital buffers, which have returned to historic levels, may not provide much room to meet strong credit demand as the economy recovers.
  - Parts of the financial system remain vulnerable to a modestly adverse scenario.
- Interlinkages and conditional risk:
  - In a modestly adverse scenario the banking sector could face further difficulties; vulnerabilities concentrated in regional and smaller banks could be amplified via interlinkages (including foreclosures and real estate price effects).
  - Capital injections substantially lowered individual institutions’ contingent liabilities and reduced systemic tail risk, but portfolio clean-up may take time.

### Scenarios, methodology, and limitations
- Core scenarios:
  - Baseline: consistent with the IMF’s April 2010 World Economic Outlook (WEO).
  - Adverse: further shocks to demand and potential output plus market fears of an unsustainable fiscal situation.
- Additional scenarios:
  - Single factor shocks and alternative scenarios consistent with historical distress episodes and other FSAPs.
- Samples and caveats:
  - The three pillars analyzed slightly different samples of financial institutions (Table 1), reflecting data availability and requirements.
  - Confidence intervals are wide; findings are subject to caveats explicitly acknowledged.

### Pillar 1 — Balance-sheet based stress test: design, projections, and key quantitative findings
- Coverage and horizon:
  - Balance-sheet exercise covers 53 BHCs, representing 85.2 percent of all BHC assets; quarterly forward-projections until end-2014.
- Projection mechanics:
  - Projects revenues, losses, retained earnings to assess potential capital shortfalls over a five-year period (2010–2014).
  - Earnings recovery assumption: average annualized return on assets of 2 percent for the 2010–14 sample periods (anchored to 1990–99 average).
- Baseline scenario results:
  - Capital adequate for most banks.
  - The top four BHCs and the former broker dealers expected to maintain a 6 percent Tier 1 common equity ratio over 2010–2014.
  - Three SCAP institutions would require US$7 billion in additional capital to maintain the same ratio.
  - Subsidiaries of foreign banks would require an additional US$26 billion in capital if required to meet the same regulatory standards as domestic peers.
- Adverse scenario results:
  - Almost one third of U.S. BHCs would experience some capital shortfall.
  - U.S. BHCs would require a total of US$32 billion in additional capital to maintain a 4 percent Tier 1 common capital ratio until end-2014.
  - Almost half of this shortfall (US$15 billion) would be accounted for by three SCAP institutions.
  - Two non-SCAP regional banks would account for US$2 billion of the shortfall.
  - Ten smaller institutions would account for US$15 billion of the shortfall.
- Loss timing and peaks (selected peak loss rates and timing):
  - RRE: Baseline 2.7 2009Q4; Adverse 3.4 2011Q4; Alternative 3.5 2012Q1
  - Cons: Baseline 6.5 2010Q1; Adverse 6.5 2010Q1; Alternative 6.5 2010Q1
  - CRE: Baseline 3.4 2011Q2; Adverse 4.6 2011Q3; Alternative 5.1 2011Q4
  - C&I: Baseline 2.6 2009Q3; Adverse 2.6 2009Q3; Alternative 2.6 2009Q3
  - Other: Baseline 3.4 2009Q4; Adverse 3.8 2011Q2; Alternative 3.6 2011Q3
- Cumulative loan losses and rates:
  - Baseline cumulative loan losses expected to reach US$802 billion by end-2014 (US$592 billion for SCAP firms).
  - Total: 6.5 percent cumulative loss for 2010–11 (12.3 percent for 2010–14) as stated in text.
  - Two-year (2010–11) annual average: 3.3 percent; Multi-year (2010–14) annual average: 2.5 percent.
  - Small and regional banks two-year loss rates: 9.7 and 9.4 percent respectively (heavy CRE exposure).
- Securities write-downs (Table 4 highlights; figures in millions as presented):
  - Total for Securities: 6,932; Cumulative Losses January 2010: 296; Cumulative Loss Rate (Percent): 6.6; Share of Total (Percent): 100.0
  - Residential Mortgage: 1,472; 166; 11.3; 56.2
  - ABS (Home and Multifamily): 980; 166; 17.0; 56.2
  - Commercial Mortgage: 196; 48; 24.5; 16.3
  - Corporate: 1,115; 171; 1.5; 5.6
  - Foreign: 975; 666; 6.7; 22.2
- Life insurance sector (NAIC exercise):
  - Largest 30 life insurers (accounting for 68 percent of U.S. life insurance premium income).
  - Aggregate RBC ratio would decline from 906 percent (as of end-2009) to 521 percent under adverse scenario.
  - Under the scenario, 5 out of the 30 companies would have RBC below 300 percent.

### Pillar 1 — Capital metrics, assumptions, and shortfall estimates
- Capital metrics:
  - (i) Tier 1 capital to risk-weighted assets with 6 and 8 percent thresholds;
  - (ii) SCAP’s tier 1 common capital/risk-weighted assets ratio with 4 and 6 percent thresholds;
  - (iii) Tangible common equity to tangible assets ratio with 4 and 6 percent thresholds.
- Aggregate system shortfall estimates (selected):
  - Three SCAP institutions would require US$7.4 billion to maintain a 6 percent Tier 1 common equity ratio.
  - Four regional banks may require US$1.3 billion.
  - Seven smaller institutions would likely require US$6.3 billion.
  - Subsidiaries of foreign banks may require up to US$26.3 billion.
  - Overall system requirement: US$40.5 billion in additional capital.
  - Top 4 institutions would need to raise US$40.4 billion if required to maintain a 5.9 percent tangible common equity to tangible assets ratio.
- Two-year sensitivity:
  - System could require as much as US$33.6 billion in additional capital to maintain a 6 percent tier 1 common capital ratio (most borne by foreign banks: US$26.3 billion).

### Pillar 2 — Distress-dependence, SMFST framework, and main results
- Purpose and methodology:
  - Systemic macro-financial stress test (SMFST) interprets the financial system as a portfolio of FIs; uses market-based CDS data adjusted for risk aversion and CIMDO to recover portfolio multivariate density (PMD).
  - Eight-step procedure: scenario definition; treat system as portfolio of FIs; infer PoDs; adjust for risk aversion; model PoDs as functions of macro/financial variables; model PMD; simulate systemic losses and contributions; estimate stability measures and spillovers.
- Sample coverage in SMFST:
  - Includes GS, MS, BoA, C, JPM, WFC, STI, USB, COF, PNC, MET, AIG, FNM, FRE (cover approximately 78 percent of depository institutions’ total assets in 2009Q3).
- Main results and dynamics:
  - Expected losses (ELs) likely to decline from 2008 peak under both scenarios; tail risks (ULs) remain substantial and above end-2007 levels through the horizon.
  - Considerable interconnectedness and spillovers identified.
  - Correlation between size (total assets) and marginal contributions to systemic risk was 0.6–0.8 in 2008–2009.
- Global extensions:
  - Banking Stability Index for the global system remains at levels similar to August 2008.
  - On average, PoDs of U.S. banks remain higher than those of European and Asian banks.
  - Tight interlinkages persist between U.S. and European banks; conditional probabilities of distress eased from 2009 peaks but remain significant.
- Key systemic loss tables (Table 10 & 11 highlights; values preserved):
  - Systemic Expected Losses (Bn US D): 2007 280.2; 2008 1251.0; 2009 820.6; 2010 750.6; 2011 340.3; 2012 200.2; 2013 210.2
  - Systemic Extreme Losses (VaR 99% Bn USD): 2007 1821.5; 2008 4273.3; 2009 3302.5; 2010 3262.5; 2011 2241.7; 2012 1911.5; 2013 1921.5

### Pillar 2 — Caveats and interpretation limits
- Not forecasts; “what if” scenario outcomes only.
- Performed exclusively with publicly available information; no supervisory exposures or actual interconnections data were available.
- PoDs inferred from historical CDS co-movements and adjusted for risk aversion; market liquidity, policy interventions, and structural changes can influence estimates.

### Pillar 3 — Systemic Contingent Claims Analysis (Systemic CCA): approach and findings
- Objective and inputs:
  - Extend CCA to a multivariate portfolio to measure market-implied contingent liabilities and joint tail risk using equity and CDS data for 36 institutions (January 3, 2007–January 29, 2009 daily sample and extensions).
- Measurement concepts:
  - Implicit put option PE(t) valued from implied asset A, asset volatility σA, default barrier B, horizon T, discount rate r.
  - α(t) = 1 − (PCDS(t) / PE(t)) measures fraction of total potential loss covered by implicit guarantees.
- Historical and peak market-implied contingent liabilities:
  - Market-implied joint contingent liabilities peaked at about US$140 billion at the end of March 2009, averaging US$74 billion over the sample period.
  - Joint tail-risk (95th percentile expected shortfall) exceeded US$1 trillion in April 2008 and almost reached US$3 trillion in October 2008.
- Sample and summary statistics (Table 14 preserved; units US$ billions):
  - April 1, 2007 - Jan. 29, 2010: 50th percentile 75; 95th percentile 144; ES (95%) 336
  - Pre-Crisis: 50th percentile 61; 95th percentile 922; ES (95%) 46
  - Crisis Period 1: 50th percentile 48; 95th percentile 315; ES (95%) 932
  - Crisis Period 2: 50th percentile 12; 95th percentile 1170; ES (95%) 290
  - Crisis Period 3: 50th percentile 86; 95th percentile 145; ES (95%) 260

### Systemic CCA — Scenario projections (Table 17 and narrative)
- Forecasting Period 2010 Q1–2014 Q4 (Table 17 averages, in billion US dollars):
  - Baseline Scenario:
    - Market-Implied Contingent Liabilities: 50th percentile 31; VaR (95%) 92; ES (95%) 180
    - Market-Implied Expected Losses: 50th percentile 75; VaR (95%) 219; ES (95%) 429
  - Adverse Scenario:
    - Market-Implied Contingent Liabilities: 50th percentile 41; VaR (95%) 130; ES (95%) 382
    - Market-Implied Expected Losses: 50th percentile 97; VaR (95%) 308; ES (95%) 910
- Projection highlights:
  - Baseline: Median implied contingent liabilities ~US$31 billion; 90th percentile >US$60 billion.
  - Adverse: Median implied contingent liabilities ~US$41 billion; 90th percentile over US$120 billion in 2011.
  - Systemic tail risk (ES95) in baseline for expected losses peaks at US$591 billion in 2012 Q2; adverse scenario ES95 exceeds US$3 trillion in 2011 Q4.

### Systemic CCA — Counterfactuals and policy insights
- Counterfactual capital injection simulations:
  - Capital injections into largest TARP recipients significantly lowered individual contingent liabilities and systemic tail risk; doubling injections had little additional effect.
  - Without capital support, average market-implied contingent liabilities from banking sector would have increased by more than 50 percent in 2009 (95 percent VaR).
- Fair-value systemic surcharge (Appendix X):
  - Estimated average annual systemic surcharge for systemically important financial institutions:
    - At least 49 basis points (April 1, 2007 - Jan. 29, 2010 50th percentile: Systemic Cont. Liabilities 74; annual fee 49).
    - 39 basis points if based on July 1, 2007-Sept. 15, 2008 (50th percentile: 59; annual fee 39).
  - Table 21 summary (selected rows preserved):
    - April 1, 2007 - Jan. 29, 2010: 50th percentile 74; annual fee 49; 95th percentile 214; annual fee 142
    - Pre-Crisis: 50th percentile 59; annual fee 39; 95th percentile 91; annual fee 60

### Macro-financial linkages, dynamic factor modeling, and scenario inputs
- Macro-financial link:
  - Monthly implicit put option values for 36 firms (01/03/2007–01/29/2010) fed into a multivariate dynamic factor model with macro variables as exogenous covariates.
  - Macro variables used include: nominal and real GDP growth, real consumption, output gap, unemployment rate, housing prices, ROA, and three-month-LIBOR–T-bill spread.
- Selected macro assumptions (Table 18 excerpts; series preserved exactly)
  - Baseline scenario (percent change, unless otherwise noted): Real GDP: 3.1 2.6 2.4 2.5 2.4; Unemployment rate (percent): 9.8 8.9 7.0 5.8 5.5; Return on assets (annualized; percent): 1.7 1.8 1.8 1.8 1.9
  - Adverse scenario series preserved exactly in source.
- Loan loss and house price modeling:
  - CRE price drop since mid-2006: about 30 percent; CRE loan maturities estimated by Congressional Oversight Panel at about US$1.4 trillion maturing in 2010–14 with nearly half seriously delinquent or “underwater.”
  - House prices baseline: rise over forecast but remain 24 percent below 2006:Q2 peak by end-2014.

### Policy implications and recommendations (extracted from results)
- Encourage weak financial institutions to raise capital: organic growth unlikely to eliminate shortfalls.
- Build interagency and system-wide stress testing capabilities to capture dependencies and spillovers (SCAP experience highlighted).
- Use market-based measures (CCA/SMFST) to quantify government contingent liabilities and support design of systemic risk surcharges (counter-cyclical design possible).
- Recognize limitations of market-implied measures: adjust CDS-derived PoDs for risk aversion and account for potential distortions from guarantees and liquidity.

### Data, sample coverage, and methodological notes (selected)
- Balance-sheet exercise based on publicly available information as of end-March 2010.
- Balance-sheet based exercise sample: 53 BHCs covering 85.2 percent of BHC assets with sub-sample asset shares: top 4 46.7 percent; 2 former investment banks 10.3 percent; 9 regional banks 7.7 percent; 3 processing banks 2.7 percent; 3 consumer banks 3.2 percent; 21 “small” banks 3.4 percent; 11 foreign banks 11.1 percent; residual 14.8 percent.
- Systemic CCA sample: 36 institutions including 17 SCAP BHCs, major broker-dealers, GSEs (Fannie Mae and Freddie Mac until conservatorship on September 8, 2008), AIG, and 8 other insurance groups; daily data January 3, 2007–January 29, 2009 for the primary analysis.

*Source: _cr10244 - References (IMF staff estimates, SNL Financial, Bloomberg, and related appendices as presented in the supplied content).*

### References .............................................................................................................

### _cr10244 - References

### Three main pillars of the stress testing analysis
- Balance-sheet based macroprudential stress tests (Section II)
  - Used publicly available financial statements and macroeconomic data to forecast financial firms’ capital needs.
  - Modeled how macroeconomic developments may affect the health of financial institutions and their lending capacity.
  - Did not account for default dependencies across institutions; may underestimate tail-risk (Figure 1).
- Macroprudential stress testing exercise with distress (Section III)
  - Accounted explicitly for distress dependencies across financial institutions using market-based credit default swap (CDS) data.
  - Computed measures of probability of default and provided estimates of unexpected losses, interconnectedness and spillovers.
  - Noted limitation: in an environment where few uninsured creditors bore the burden of financial distress, CDS data may not fully capture true probability of default.
- Estimates of government’s potential contingent liabilities implied by financial market prices (Section IV)
  - Used equity and CDS prices in a high-dimensional extension of contingent claims analysis (CCA) to capture risk-adjusted balance sheets and interdependencies.
  - Focused on expected shortfall: average density of extreme losses beyond the 95 percent Value-at-Risk (VaR) (Figure 1).

### Key findings and system-wide assessment
- The stress tests relied on publicly available information and models subject to considerable estimation uncertainty; lack of granular supervisory information was a constraint.
- Under the baseline macro scenario, following substantive public and private capital injections, equity buffers appear satisfactory for the system as a whole.
- Stress tests indicate vulnerabilities:
  - Some individual institutions, including a few of the smaller SCAP institutions, may be less well-positioned to absorb future losses through earnings.
  - Existing capital buffers, which have returned to historic levels, may not provide much room to meet strong credit demand as the economy recovers.
  - Parts of the financial system remain vulnerable to a modestly adverse scenario.
- Importance of macro-financial linkages and dependencies among the largest institutions:
  - Analysis suggests that in a modestly adverse scenario the banking sector could face further difficulties.
  - Vulnerabilities concentrated in regional and smaller banks that could be amplified via interlinkages (including foreclosures and real estate price effects).
  - Capital injections substantially lowered individual institutions’ contingent liabilities and reduced systemic tail risk, but portfolio clean-up may take time.

### Scenarios, methodology, and limitations
- Two core scenarios were used in all tests:
  - Baseline: consistent with the IMF’s April 2010 World Economic Outlook (WEO).
  - Adverse: predicated on further shocks to demand and potential output, plus impact of market fears of an unsustainable fiscal situation and related inflationary expectations.
- Single factor shocks and alternative scenarios were also considered; magnitudes were consistent with historical distress episodes and with ranges analyzed in other FSAPs.
- The three pillars analyzed slightly different samples of financial institutions (Table 1), reflecting data availability and requirements.
- Confidence intervals are wide; findings are subject to caveats explicitly acknowledged in the relevant sections.

### Complementary components and institutional developments
- Two related components complemented the stress testing:
  - A survey of authorities’ own stress testing practices (Appendix I).
  - A detailed stress test for life insurance companies carried out with the National Association of Insurance Commissioners (NAIC) (Appendix II).
- Authorities signaled intention to conduct periodic forward-looking scenario analyses to enhance understanding of adverse changes in the operating environment for firms and the system.
- SCAP experience illustrated benefits of building interagency and system-wide stress testing capabilities; authorities are undertaking broader horizontal (cross-institution) reviews.

### Technical focus and risk metrics highlighted
- Figure 1 emphasized conceptual differences in loss measurements across pillars:
  - Expected Loss (EL), Unexpected Loss (UL), VaR95%, VaR99%, and ES95% (average density beyond VaR95%).
- Pillar 3 centered on expected shortfall (ES95%) as the measure of tail risk.
- Market-based measures (CDS, equity prices) were used to compute probabilities of distress, unexpected losses, and market-implied contingent liabilities.

*Source: _cr10244 - References.*

### 8. In the first pillar, i.e., the balance sheet-based macroprudential analysis, the

### 8. In the first pillar, i.e., the balance sheet-based macroprudential analysis, the

### Pillar 1 — Balance-sheet based stress test design and scope
- The team stress tested a wide set of large bank holding companies (BHCs) to gauge the soundness of the banking system as a whole, and explore differences across peer groups.
- The analysis projects revenues, losses, and retained earnings to assess potential capital shortfalls over a five-year period.
- Consideration was given to firm-specific differences in earnings and losses, based on portfolio composition and historical performance.
- The analysis attempted to account for deleveraging, de-risking, asset on-boarding, and impaired securitization on BHC system-wide asset growth.
- Some features could not be accounted for due to data and modeling constraints (example given: purchasing accounting assumptions on acquired assets).

### Pillar 1 — Key quantitative findings (balance-sheet based results)
- Under the baseline scenario:
  - Capital would be adequate for most banks.
  - The top four BHCs and the former broker dealers are expected to maintain a 6 percent Tier 1 common equity ratio over 2010–2014.
  - Three SCAP institutions would require US$7 billion in additional capital to maintain the same ratio.
  - Subsidiaries of foreign banks would require an additional US$26 billion in capital if required to meet the same regulatory standards as domestic peers.
- Under the adverse scenario:
  - Almost one third of the U.S. BHCs would experience some capital shortfall.
  - U.S. BHCs would require a total of US$32 billion in additional capital to maintain a 4 percent Tier 1 common capital ratio until end-2014.
  - Almost half of this shortfall (US$15 billion) would be accounted for by three SCAP institutions.
  - Two non-SCAP regional banks would account for US$2 billion of the shortfall.
  - Ten smaller institutions would account for US$15 billion of the shortfall.
- Assumptions underpinning these results include:
  - Residential real estate and commercial real estate losses continue to rise until 2011.
  - Losses on consumer loans start to decline from their 6.5 percent peak in the first quarter of 2010.

### Pillar 1 — Sensitivities, vulnerabilities, and interpretation
- The results illustrate high sensitivity of BHCs’ asset quality and capital positions to developments in the housing sector and the broader economy.
- There is much uncertainty about banks’ earnings outlook and the shape and height of loss profiles; losses are expected to be a drag on retained earnings and credit growth for some time.
- Identified fragilities:
  - Regional and smaller institutions present fragilities that do not appear systemic but could hamper economic recovery in local communities and have broader repercussions on bank loss rates.
  - Low capitalization of foreign-owned BHCs is a potential macroprudential vulnerability; a withdrawal of parental support could trigger sharp domestic exposure retraction.
- Conclusion: despite strong recapitalization efforts, it will take time to clean up banks' balance sheets.

### Liquidity and funding risks (market liquidity, rollover risk)
- Market liquidity risks appear to have declined for the financial system as a whole, although financial firms remain vulnerable to funding rollover risk.
- With short-term liquidity infusion, financial institutions improved liquidity buffers but shortened funding maturity profiles.
- Financial firms face rollover risks arising from a bunching of assets maturing in 2011–13.
- Lack of supervisory data limited the analysis; strains could be exacerbated if BHCs cannot refinance maturing loans, leading to deterioration in CRE and residential real estate losses.

### Life insurance sector stress tests (linked but separate exercise)
- Separate stress tests were carried out for the largest 30 life insurance companies (accounting for 68 percent of U.S. life insurance premium income) in cooperation with NAIC.
- Adverse scenario combined negative asset shocks, a liability-side shock impacting variable annuity writers, and a major insurance shock (a pandemic).
- Aggregate risk-based capital (RBC) ratio would decline from 906 percent (as of end-2009) to 521 percent.
- Under the scenario, 5 out of the 30 companies would have RBC below 300 percent.
- Companies with substantial variable annuity business would be particularly hard hit, but no company would have a negative RBC under the scenario.

### Pillar 2 — Distress-dependency (interdependencies among major financial firms)
- The system was tested for distress dependencies among major financial firms using a forward looking, market data-based framework.
- Interdependencies are assessed via a statistical model using market-based data; results are subject to uncertainty and do not represent direct lending or counterparty exposure data.
- Focus was on losses defined as the value of defaulted loans less recoveries.
- Key findings:
  - Expected losses likely to decline from the 2008 peak under both scenarios, reflecting improving macroeconomic developments.
  - Tail risks remain substantial under both scenarios; systemic unexpected losses incorporating interconnectedness are likely to remain at elevated levels in the near future.
  - Considerable interconnectedness and spillovers were identified.
  - The marginal contribution of an individual firm to systemic risk depends on size and linkages; correlation between size (total assets) and marginal contributions was 0.6–0.8 in 2008 and 2009.
  - Measures of interlinkages between banks and non-financial corporates declined from the first quarter of 2009 but remain significant and appear to be increasing more recently.
  - The identified linkages derive from a statistical model and do not represent actual lending, counterparty exposures, or other common exposures.

### Pillar 2 — Global context and Banking Stability Index
- An extension shows global vulnerabilities eased from recent highs, although systemic tail risk remains elevated.
- The team’s Banking Stability Index for the global system remains at levels similar to those observed in August 2008.
- The index trend is consistent with average probabilities of default observed by region.
- On average, probabilities of default of U.S. banks remain higher than those of European and Asian banks.
- Tight interlinkages persist between U.S. and European banks, implying ongoing cascade risk.
- Conditional probabilities of distress between U.S. and European banks eased from first-quarter-2009 peaks but remain significant.
- The probability of problems at large U.S. banks spilling over to other global banks was appreciably high as of December 2009.
- Some U.S. banks appear vulnerable to negative developments at other global banks or sovereigns.

### Pillar 3 — Systemic Contingent Claims (market-implied government contingent liabilities)
- The Systemic CCA framework estimated the financial market’s expectation of government contingent liabilities using daily data for 36 financial firms in 2007–2009.
- Findings:
  - More than half of total expected losses—as indicated by lower default risk implied by CDS spread compared to equity prices—could have become public sector liabilities.
  - Market-implied joint contingent liabilities peaked at about US$140 billion at the end of March 2009, averaging US$74 billion over the sample period.
  - Market perception of implicit or explicit guarantees depresses CDS prices, which limits systemic risk measures based on CDS-implied probabilities of default to the retained risk in the financial sector.
- Joint tail-risk measure (95th percentile expected shortfall) findings:
  - Market-implied contingent liabilities exceeded US$1 trillion in April 2008 and almost reached US$3 trillion in October 2008.
  - Housing GSEs were large contributors to systemic risk up to conservatorship.
  - BHCs needing additional capital according to SCAP contribute far more to systemic tail risk than other SCAP firms after the Lehman Brothers collapse, especially if capital need is estimated jointly.
  - Simulations indicate capital injections into the three largest TARP recipients significantly lowered individual contingent liabilities and systemic tail risk.

### Data, sample coverage, and methodology notes
- The balance-sheet based exercise covers 53 BHCs, representing 85.2 percent of all BHC assets (sample grouped into 7 sub-categories).
- Sub-sample composition (percent of sample assets): “top 4” account for 46.7 percent; 2 former investment banks account for 10.3 percent; 9 regional banks account for 7.7 percent; 3 processing banks account for 2.7 percent; 3 consumer banks account for 3.2 percent; 21 “small” banks account for 3.4 percent; 11 foreign banks account for 11.1 percent; residual category accounts for 14.8 percent.
- The exercise is based entirely on publicly available information as of end-March 2010.
- The stress tests forecast key balance-sheet elements to capture interactions between earnings potential, capital positions, and loss absorption capacity, with implications for lending and vulnerability to targeted shocks.
- Results are point estimates and subject to uncertainty from statistical model specification, bank-level detail limitations, potential more severe unforeseen events, and assumptions on banks’ future business practices.
- Earnings recovery assumption: average annualized return on assets of 2 percent for the 2010–14 sample periods, anchored to 1990–99 historical average.
- Identified upside and downside risks:
  - Downside: regulatory reforms, greater risk retention, lower credit growth, strategic defaults, depressed collateral values, uncertain recovery rates.
  - Upside: banks’ ability to raise private capital or reduce dividend policy.

*Source: _cr10244 - 8. In the first pillar, i.e., the balance sheet-based macroprudential analysis, the*

### 24. The assessment of BHCs’ capital adequacy over the forecast period employed

### 24. The assessment of BHCs’ capital adequacy over the forecast period employed

### Capital metrics and thresholds
- Three capital metrics were used:
  - (i) the ratio of tier 1 capital to risk-weighted assets with 6 and 8 percent thresholds;
  - (ii) the SCAP’s tier 1 common capital/risk-weighted assets ratio with 4 and 6 percent thresholds;
  - (iii) a tangible common equity to tangible assets ratio with 4 and 6 percent thresholds.
- Historical context: SCAP institutions maintained an average 10 percent tier 1 capital ratio during the crisis and their tier 1 common capital ratio was on average 7.4 percent over 1997–2007 (Figure 2).
- Assumptions regarding dividend and capital behavior:
  - Baseline: banks would not raise capital over the sample horizon and profit-making financial firms would not reduce their dividends policy in anticipation of a future capital need.
  - Adverse scenario: banks were expected not to pay out common stock dividends, in line with the authorities’ SCAP exercise.

### Data, sample, and projection horizon
- Sample and data sources:
  - Realized quarterly data from end-2007 to end-March 2010; quarterly forward-projections until end-2014.
  - Bank-specific data: publicly available Y-9C reports filed with the Federal Reserve, obtained from SNL Financial’s database.
  - Augmenting data: SEC data for non-banks before BHC conversion in 2008, Bloomberg for capital raising measures and securities write-downs, and the U.S. Treasury Department’s website www.FinancialStability.gov for TARP repayments and dividends.
- The exercise spans a seven-year horizon and produced quarterly forward projections until end-2014.

### Framework for baseline scenario and macro-financial linkages
- Projection elements:
  - Projected firms’ net revenues, losses, and balance sheet expansion used to assess potential capital shortfalls over a five-year period.
  - A bank’s capital shortfall computed based on its lowest capital position over the horizon.
  - Consideration of firm-specific differences in earnings and losses based on portfolio composition and historical performance.
  - Specific post-crisis factors incorporated: deleveraging and de-risking efforts, FAS 166/167 on-boarding of previously off-balance-sheet assets, and greater risk retention due to impaired securitization.
  - Incorporation of firms’ ability to accumulate tax assets in loss-making quarters to offset future tax liabilities.
- Macro-financial linkages:
  - Macro variables drive financial variables: nominal GDP growth drives asset growth; loan loss rates reflect real GDP, real consumption, unemployment, and the output gap.
  - Lending standards and house prices are forecasted separately (Appendix III).
  - Judgmental adjustments may be needed as there is no universal consensus on macro-financial relationships.
- Baseline macro assumptions (from IMF’s April 2010 World Economic Outlook):
  - Output gap closed over the medium term from a negative level in 2009.
  - Unemployment rate remaining elevated (above 8 percent) until end-2011 before dropping to 5 ½ percent by end-2014.
  - Real GDP growth expected to peak at 3.1 percent in 2010 and to stabilize around 2.5 percent by 2012.
  - House prices expected to rise over the forecast horizon, peaking at 4.1 percent in 2011.

### Loan loss modeling and assumptions
- Five categories of loan charge-off rates estimated industry-wide from regression analysis: CRE, RRE, C&I, consumer (CONS), and “other” (other computed as simple average of the other four).
- BHC-specific charge-off projection formula (notation preserved): ,j iq COR where i denotes firms, j indexes loan type, q denotes time. Forecasted rates computed recursively taking as base previous quarter’s value and applying industry-wide change ,j Iq COR: ,,,1 j jj Iq iqiq CORCORCOR  
- No adjustment made for stricter underwriting standards post-2009; omission could lead to upward bias in outer years.
- Limited adjustments for 2008 mergers and acquisitions; unlike SCAP, no purchase accounting adjustment (SCAP reduced estimated losses by US$64 billion).
- Minor adjustments for investment banks converted to BHCs in late 2008 (e.g., Morgan Stanley’s 2008Q4 loan loss rates omitted from moving average; earnings path adjusted toward average fixed-effects of top six firms).

### Timing and peaks of loss rates (Table 2)
- Peak loss rates by loan type and scenario (Percent):
  - RRE: Baseline 2.7 2009Q4; Adverse 3.4 2011Q4; Alternative 3.5 2012Q1
  - Cons: Baseline 6.5 2010Q1; Adverse 6.5 2010Q1; Alternative 6.5 2010Q1
  - CRE: Baseline 3.4 2011Q2; Adverse 4.6 2011Q3; Alternative 5.1 2011Q4
  - C&I: Baseline 2.6 2009Q3; Adverse 2.6 2009Q3; Alternative 2.6 2009Q3
  - Other: Baseline 3.4 2009Q4; Adverse 3.8 2011Q2; Alternative 3.6 2011Q3
- Observations:
  - Consumer loans reached 6.5 percent at end-January 2010.
  - Residential real estate rose to 2.7 percent at end-2009.
  - CRE losses expected to peak later (until mid-2011).
  - C&I loss rate peaked at 2.6 percent at end-September 2009.

### Cumulative losses and distribution (Table 3 and text)
- Baseline cumulative loan losses expected to reach US$802 billion by end-2014 (US$592 billion for SCAP firms).
- Cumulative loss rates:
  - Total: 6.5 percent cumulative loss for 2010–11 (12.3 percent for 2010–14) as stated in text; Table 3 reports variant figures across groups and scenarios (see Table 3 for group-level detail).
- Two-year and multi-year annual averages:
  - Two-year (2010–11) annual average: 3.3 percent for 2010–11.
  - Multi-year (2010–14) annual average: 2.5 percent for 2010–14.
- Group-specific two-year loss rates (2010–11) noted in text:
  - Consumer banks face the largest two-year loss rate.
  - Small and regional banks face 9.7 and 9.4 percent, respectively (reflecting heavy CRE exposure).
- Comparison with SCAP:
  - Two-year loss rates are below the 9.1 percent 2009–2010 loss rate assumed in the SCAP stress test.

### Securities write-downs (Table 4)
- Methodology: declines in market valuations using methodology from recent Global Financial Stability Reports; under the baseline, no additional AFS write-downs and no shocks to marked-to-market trading account securities.
- Historical realized write-downs: BHCs reported cumulative US$385 billion of realized marked-to-market securities write-downs since end-2007; model estimated US$296 billion.
- No allowance made for write-ups to banks’ securities holdings.
- Table 4 figures (Estimated Holdings January 2010; Cumulative Losses January 2010; Cumulative Loss Rate (Percent); Share of Total (Percent)):
  - Residential Mortgage: 1,472; 166; 11.3; 56.2
  - Agency (Prime Conforming): 492; 00; 0.0; 0.0
  - ABS (Home and Multifamily): 980; 166; 17.0; 56.2
    - of which: Non-agency Prime MBS: 530; 00; 0.0; 0.0
    - of which: ABS (CDOs, other MBS): 450; 166; 37.0; 56.2
  - Consumer: 142; 00; 0.0; 0.0
  - Commercial Mortgage: 196; 48; 24.5; 16.3
  - Corporate: 1,115; 171; 1.5; 5.6
  - Governments: 580; 00; 0.0; 0.0
  - Foreign: 975; 666; 6.7; 22.2
  - Total for Securities: 6,932; 296; 6.6; 100.0
- Sources for Table 4: Bloomberg and IMF staff estimates.

### Real estate and systemic risk observations
- CRE outlook:
  - CRE losses projected to remain high unless commercial property prices recover from a 30 percent fall since mid-2006.
  - Congressional Oversight Panel (2010) estimated about US$1.4 trillion in loans will mature in 2010–14, nearly half already seriously delinquent (90 days or more past due) or “underwater.”
- RRE vulnerabilities:
  - As of end-March 2010, seriously delinquent loans (90-days or more overdue) accounted for 5 percent of total mortgages; actual foreclosures accounted for 1.25 percent of total loans.
  - Rising gap between delinquencies and foreclosures indicates pent-up supply and potential downward pressure on house prices.
  - Loan modifications can mitigate immediate foreclosures but may lead to high re-default rates, postponing losses.

### Earnings model and key regression results (ROA)
- Focus: pre-provision, pre-tax, and pre-dividend net revenues as a percentage of total assets (return on assets, ROA).
- Regression specification (fixed-effects panel, 53 BHCs):
  - 1ittititiit ROAXyzu       
  - Explanatory variables: vector X t of three macro variables (real GDP quarterly growth rate, output gap, lagged quarterly unemployment growth rate), bank-specific lagged loan-to-asset ratio y it-1, and financial market variable z it (3-m Libor to 3-m TBill spread). u it is unit-specific residual; ε it is residual.
- Data estimation period: 1990Q1–20010Q1 using quarterly frequency data; macro variables seasonally adjusted.
- Final model specification highlighted in Column (4) of Table 5. Selected coefficient estimates (Column (4), with significance):
  - Loan-to-asset (lagged): 0.307** (robust p-values indicated; ** p<0.05)
  - Real GDP quarterly growth: 1.687** (p-values indicated)
  - 3-m Libor to 3-m TBill: -0.150*** (*** p<0.01)
  - Output gap: 0.0197*** (*** p<0.01)
  - Dummy for 2007–2008: -0.0929*** (included in other columns)
  - Constant: 0.393*** 
- Model statistics (as reported in Table 5):
  - Number of observations for fixed effects regressions: 3,401
  - R-squared and related metrics vary by specification (e.g., macro data R-squared 0.650; fixed effects within R-square 0.138).
- Forecast implementation:
  - Estimated coefficients applied to forecasted explanatory variables.
  - Resulting retained earnings fed back into total assets each quarter.
  - Fixed effect captures bank-specific differences; no sub-group modeling attempted.

*Source: SNL Financial; IMF staff estimates.*

### 40. A wide range of model specifications were tested to estimate earnings, including

### 40. A wide range of model specifications were tested to estimate earnings, including

### Model design, choice of explanatory variables, and caveats
- Explanatory variables were restricted to those directly linkable to the macroeconomic model used for scenario analysis or for which there was an in-house forecasting model (examples: GDP, output gap, unemployment, real consumption, house prices, yields on the London Interbank Offered Rate (LIBOR) or treasury bills).
- Real personal consumption expenditures growth and house prices (both unadjusted and detrended) were found to be statistically insignificant.
- The lagged log of total assets (L.lnta) was not statistically significant.
- Different financial market spreads yielded limited differences; the three-month Libor to treasury bill spread was chosen for convenience as it was part of the macroeconomic forecast.
- The exercise has a macro-prudential focus and is not designed as a definitive forecast for individual institutions or business lines. Estimates may be biased and have significantly reduced accuracy in outer forecast years.
- A 90 percent confidence interval could yield quarterly return on asset estimates anywhere between 0.25-0.75 percent.

### Statistical methods and specification selection
- Model selection used a range of statistical tests:
  - Akaike and Bayesian Information Criteria to narrow explanatory variables.
  - Hausman specification test and likelihood-ratio test to differentiate specifications.
  - Durbin-Watson d-statistic and Breusch-Godfrey Lagrangian multiplier test indicated serial correlation would disappear with inclusion of bank-specific variables.
  - Random effects models were consistently rejected by the Breusch and Pagan Lagrangian multiplier test.
  - Multi-level mixed effects panel models were also run.
- Crisis dummies (2007–2008 downturn) were tested but omitted from final specification because real GDP was judged to capture crisis effects.

### Earnings projections (ROA) — baseline and adverse scenarios
- Industry-wide projected return on asset is expected to average 1.96 percent on an annualized basis over the forecast horizon.
- Comparison to other exercises:
  - SCAP assumed banks’ return on assets would remain almost 15 percent below the past twenty-year average for 2009–10, or around 1.6 percent on an annualized basis.
- By bank category, projected annualized ROA over the forecast horizon:
  - Top 4 bank holding companies: 2.17 percent (around 1990-99 historical average of 2.2 percent).
  - Regional banks: 2.33 percent.
  - Small banks: 1.75 percent.
  - Foreign banks: 1.36 percent.
- Historical pre-crisis comparison (years immediately preceding the crisis):
  - Regional banks: 2.37 percent.
  - Small banks: 2.26 percent.
  - Foreign banks: 1.70 percent.
- Near-term dynamics:
  - Projected ROA slightly lower over the next two years: 1.9 percent for the system relative to 2 percent over 2010–14.
  - Adverse scenario: system expected to average an annualized 1.6 percent until end-2012 or 1.7 percent over the forecast horizon.

### Retained earnings, tax treatment, and deferred tax assets (DTAs)
- Tax framework:
  - A simple 30 percent flat tax rate applied on banks’ corporate income.
  - Accounting for carry-backs: operating losses could be carried back for two years to recover income taxes previously paid; these carry-backs referred to as deferred tax assets (DTAs).
  - Up to 10 percent of Tier 1 capital assumed as allowed DTAs (per U.S. BHC prudential requirements).
- DTA outcomes:
  - Cumulative DTAs peaked at end-March 2009 at US$143 billion for the system.
  - 68 percent of cumulative DTAs at peak were accounted for by the top 4 institutions.
  - By end-2014, DTAs would help reduce future tax liabilities by 82 percent.
  - Under the baseline, 21 institutions would not have to pay income tax over the sample horizon, including one of the top 4 institutions and 5 regional banks.
  - In the adverse scenario, the number of firms not paying income tax would rise to 31 institutions, including 3 of the top 4 institutions.
- Dividend rule assumptions:
  - When net after-tax income positive, assumed annualized dividend rates: 5 percent for TARP preferred shares, 8 percent for other preferred shares (relative to an average of 5 percent over 1990–99), and 15 percent for common equity (relative to an average of 22 percent over 1990–99).
  - Resulting averages: 11.6 percent annualized average dividend rate for common equity and 2.6 percent annualized average dividend rate for preferred shares.
  - In downside risk scenarios, banks were not expected to pay common share dividends.
- Retained earnings projections:
  - Industry retained earnings (PPNR minus loan charge-offs, securities write-downs, taxes, and dividends) would remain positive but low (slightly above $20 billion on average) until end-2011, then start rising.
  - Regional banks: average quarterly retained earnings of less than US$1 billion until end-2011.
  - Small banks: negative retained earnings until the first quarter of 2012.
  - Over 2010–14 forecast horizon, retained earnings for the system would average US$43 billion on a quarterly basis (US$34 billion for SCAP firms, US$4 billion for regional banks, and less than US$1 billion for small banks).

### Balance sheet expansion and asset dynamics
- Factors expanding total system assets over 2010–2014:
  - Weak securitization markets predicted to add US$195 billion to total system assets.
  - Introduction of FAS 166/167 accounting rules in 2010 assumed to expand balance sheets by US$375 billion.
  - Retained earnings added back into total assets, adding US$670 billion over the sample horizon (64 percent of retained-earnings-driven expansion generated by the top 6 firms).
- Factors reducing asset growth:
  - Asset sales subtracted US$375 billion.
  - Asset maturities without rollovers reduced assets by US$496 billion.
- Distribution:
  - Except for retained earnings (estimated bank-by-bank), balance sheet expansion factors were distributed across firms according to their share of total system assets.
- Loan portfolio and loan-to-asset dynamics:
  - Banks allowed to grow loans in proportion to asset growth with a constant loan-to-asset ratio, plus permission to expand loans by drawing down up to 5 percent of “other assets” for 8 consecutive quarters (or until “other assets” reached 20 percent of total assets).
  - As a result, the loan-to-asset ratio was raised by 7 percentage points to 50 percent by the end of the forecast horizon.

### Risk-weighted assets and compositional adjustments
- Since end-2007, the ratio of risk-weighted assets to total assets had fallen by over 5 percentage points to 61 percent (the lowest point recorded since introduction of RWAs).
- Baseline assumption: risk-weighted to total asset ratios would return progressively back to their 2000–2005 average by mid-2011.
- Adverse scenario assumption: RWA/TA ratio would remain constant at the low end-March 2010 level.
- BHC balance sheets allowed to adjust composition to maximize room for credit expansion; loans permitted to grow relative to other assets per framework.

### Capital adequacy and shortfall estimates
- Baseline conclusions:
  - Industry-wide bank capital would be adequate on an industry-wide basis despite weak growth, high unemployment, and record high charge-off rates.
  - Top 4 BHCs and former broker dealers expected to maintain a 6 percent Tier 1 common equity ratio over 2010–2014.
- Identified capital shortfalls and by-group amounts:
  - Three SCAP institutions would require an addition of US$7.4 billion to maintain a 6 percent Tier 1 common equity ratio.
  - Four regional banks (including two SCAP institutions) may require US$1.3 billion in additional capital due to high CRE exposure.
  - Seven smaller institutions would likely require an additional US$6.3 billion.
  - Subsidiaries of foreign banks may require up to US$26.3 billion.
  - Overall system requirement: US$40.5 billion in additional capital.
  - Top 4 institutions would need to raise US$40.4 billion in additional capital if required to maintain a 5.9 percent tangible common equity to tangible assets ratio (or 17 times leverage).
- Two-year horizon sensitivity:
  - The capital shortfall picture does not change materially when focusing on a two-year forecast horizon.
  - System could require as much as US$33.6 billion in additional capital to maintain a 6 percent tier 1 common capital ratio, most of which borne by foreign banks (US$26.3 billion).
- Policy implication highlighted:
  - Weak financial institutions should be encouraged to raise capital, as current conditions do not allow them to grow out of their problems and organic growth is unlikely to eliminate shortfalls.

*Sources: SNL Financials and IMF staff estimates.*

### 54. The estimated capital shortfall of foreign banks is difficult to interpret. The

### _cr10244 - 54. The estimated capital shortfall of foreign banks is difficult to interpret. The

### Foreign banks and interpretative caveats
- The exercise stresses foreign institutions in the same way as domestic ones to assess the broader shock absorption capacity of the U.S. banking system.
- In normal times, foreign holding companies tend to operate with lower capital buffers than their domestic peers because they are not required to comply with U.S. regulatory capital requirements, provided their parents are deemed well-capitalized and well-managed.
- Under a global adverse shock, it could be particularly difficult for regulators to require higher capital buffers when parent banks could be equally strained.
- Although retrenchment or closure of foreign banks would likely not have systemic consequences from a financial stability perspective, it may have broader macro-prudential implications depending on the operations of the affected institutions.
- Since end-2007, foreign BHCs reduced their loan market share by 5 percentage points to 9 percent.
- Another potential vulnerability identified is the low capitalization of foreign-owned BHCs.

*Key related figure/commitment*
- Authorities intend to allocate US$30 billion of TARP money to community banks.

### Baseline scenario: credit growth and earnings (2010–2014)
- Without additional capital injections, credit growth could average around 8 percent for 2010–2014.
- Historical comparisons:
  - 1993–1996 average credit growth: 16.1 percent
  - 2004–07 average credit growth: 16.8 percent
- In the adverse scenario, average credit growth could fall by another 2 percentage points for the forecast horizon.
- Banks have ways to meet credit demand: raising new capital, curbing dividend rates, or generating higher retained earnings than anticipated.

### Adverse scenario: loan losses, retained earnings, and capital shortfalls
- Loan loss dynamics and cumulative losses:
  - Residential and commercial real estate loan losses continue to rise until 2011, peaking at 3.4 percent and 4.6 percent, respectively.
  - Cumulative loan losses expected to reach US$1.1 trillion for the system as a whole by end-2014.
  - Cumulative loss rates: 7.7 percent for 2010–11; 15.9 percent for 2010–14.
  - Institutions with securities portfolios are also expected to write-down US$100 billion of marked-to-market securities.
  - Total cumulative loss rate: 17.4 percent for 2010–14; 8.7 percent for 2010–11.
- Retained earnings:
  - Retained earnings would remain negative until 2012 for the system as a whole and until 2014 for the smaller banks.
  - SCAP firms: retained earnings turning positive by end-2011.
  - Retained earnings for the system would record an average quarterly loss of US$2.4 billion for 2010–2011 (US$1.2 billion for the regional banks and US$1.8 billion for the small institutions).
- Capital shortfalls:
  - On aggregate, U.S. BHCs would require a total of US$31.8 billion capital to maintain a 4 percent Tier 1 common capital ratio until end-2014 (US$53.6 billion including the foreign BHCs).
  - Breakdown highlights:
    - 4 regional banks would require US$8.1 billion.
    - 10 smaller institutions would require US$14.9 billion.
    - Three SCAP banks would face a shortfall of US$14.5 billion.
    - One of the top 4 institutions would need to raise US$15.2 billion to maintain a 4 percent tangible common equity to tangible assets ratio by end-2014.
  - Over the 2010–11 horizon: 10 U.S. BHCs (including two SCAP institutions) expected to face US$8.9 billion of capital shortfall to maintain a 4 percent Tier 1 common capital ratio.

### Alternative scenario (commercial real estate rollover risk)
- Rationale:
  - Market liquidity risks have declined, but FIs remain vulnerable to large volumes of commercial real estate (CRE) loans maturing between 2010 and 2014 (many with negative equity) and rising seriously delinquent mortgages.
- Assumptions and stress:
  - Macroeconomic conditions broadly similar to the Adverse Scenario for the first two years, but return faster to the baseline beyond 2011.
  - Commercial real estate prices fall by another 8 percent by end-2012 (as opposed to 3.3 percent in the Adverse).
  - House prices fall by 4.1 percent in 2010 and another 2.6 percent in 2011.
  - Banks’ difficulty rolling over maturing debt leads to higher losses on CRE loans, which peak at 5.1 percent at end-2011.
- Results:
  - Except for banks heavily exposed to CRE, macroeconomic conditions are the key determinant of banks’ financial soundness.
  - Overall, 14 U.S. BHCs would require US$20.5 billion capital to maintain a 4 percent Tier 1 common capital ratio over 2010–14 (US$7.4 billion over the 2010–11 period).
  - The results suggest that banks heavily exposed to CRE losses will find it difficult to earn their way out of problems under worse-than-expected macroeconomic conditions, but aggregate outcomes are not highly sensitive to a further small deterioration in real estate prices or recovery rates on delinquent real estate loans.
  - A broader shock to banks’ funding conditions (for example, a substantive rise in short-term spreads) would likely have a more dramatic impact on banks’ earnings and ability to absorb losses.

### Balance-sheet based conclusions and capital adequacy
- BHCs’ asset quality and capital positions are closely interlinked with developments in the housing sector and the broader macro economy.
- Loss profiles are highly uncertain but expected to drag on retained earnings and credit growth.
- Fragilities in regional and smaller institutions are not systemic but could hamper economic recovery in local communities with broader repercussions on bank loss rates.
- Despite significant improvement in BHCs’ capital buffers:
  - BHCs have almost doubled their holdings of “high-quality” capital since the crisis.
  - However, record low risk-weighted to total asset ratio, protracted high loss profiles, limited risk transfer through securitization, and recognition that financial institutions need to hold higher capital buffers than pre-crisis imply banks’ balance sheets may not be as strong as capital buffers suggest.
- Memo: Percent of total system assets listed in tables: 46.5, 10.4, 8.0, 2.8, 2.9, 3.5, 11.3, 14.6, 100.0, 88.7 (as presented in the tables).

### Macroprudential stress tests with distress dependence (SMFST)
- Purpose:
  - To assess stability of the U.S. financial system from a systemic perspective by performing a systemic macro-financial stress test (SMFST) that captures distress-dependence among financial institutions.
- Framework capabilities (quantifiable outputs):
  - Expected losses, and extreme losses (unexpected losses), accounting for distress-dependence and its changes through the economic cycle.
  - The marginal contribution of individual firms to systemic risk, reflecting both the level and the relative size of interconnectedness of each institution with the system.
  - Stability measures analyzing: evolution of tail risk in the system, distress dependence among firms, and cascade effects (the impact of distress at a given firm on other firms).
  - Spillovers between major U.S. and foreign FIs, U.S. FIs and emerging market sovereigns, and U.S. financial markets and selected U.S. non-financial corporations.
- Methodological note:
  - Calculations based on joint implementation of the Consistent Information Multivariate Density Optimizing Methodology (CIMDO) and the framework for estimating Banking Stability Measures presented in Segoviano and Goodhart (2009).

*Italic: Source — IMF staff estimates and SNL Financials, as presented in the specified chapter content.*

### 67. There are a number of caveats in interpreting the SMFST estimates:

### _cr10244 - 67. There are a number of caveats in interpreting the SMFST estimates:

### Caveats in interpreting the SMFST estimates
- The estimates are not forecasts; as in any stress test exercise, they are outcomes of “what if” calculations conducted under a baseline scenario and an adverse scenario.
- This test was performed exclusively with publicly available information. Supervisory information of banks’ portfolio compositions, asset risk parameters, off-balance sheet items, counterparty risk, and interbank exposures was not available to the FSAP team. Therefore, importantly, none of the analysis that follows benefited from actual data or information on actual interconnections (lending relationships, counterparty exposures, other common exposures) among financial institutions.
- This test was performed in the middle of considerable uncertainty in the wake of the financial crisis. While the situation has stabilized and financial conditions have improved, considerable sources of uncertainty remain for the financial system.
- The main objective of the SMFST was to assess financial stability based on systemic potential (expected and unexpected) losses and spillovers among FIs. In addition, the adequacy of existing capital buffers to withstand unexpected losses was also assessed for illustration purposes. An alternative treatment of the evolution of buffers across time is presented in Section II.
- As noted above, because supervisory data were not available, the results depend on market expectations of distress as manifested in historical CDS prices. Conclusions about future interconnections among firms are inferred from historical co-movements in CDS market prices, not from any actual data about interconnections. Market perceptions, liquidity issues in CDS markets and a series of mergers and acquisitions in the financial sector upon the crisis could influence our estimates.

### Methodology: system tested as a portfolio of FIs
- The system was tested from a systemic perspective using the systemic macro-financial stress test that interprets the financial system as a portfolio of FIs.
  - This includes the largest banks, GSEs and a large insurance company. The banks included in the exercise are Goldman Sachs (GS), Morgan Stanley (MS), Bank of America (BoA), Citigroup (C), J.P. Morgan (JPM), Wells Fargo (WFC), SunTrust (STI), U.S. Bancorp (USB), Capital One (COF), PNC, and MetLife (MET). These institutions cover approximately 78 percent of depository institutions’ total assets in 2009Q3. AIG, Fannie Mae (FNM), and Freddie Mac (FRE) were also included in the exercise.
- Interconnectedness was incorporated using a forward looking risk-based framework that allows assessment of:
  - (i) systemic potential losses,
  - (ii) the contribution of individual FIs to systemic risk,
  - (iii) financial stability measures, and
  - (iv) spillovers between U.S. and foreign FIs, U.S. FIs, and emerging market sovereigns, and between U.S. FIs and the U.S. corporate sector.

- Eight-step SMFST procedure (Figure 14):
  - Step 1: Definition of macroeconomic scenarios
  - Step 2: Conceptualization of the Financial System as a portfolio of FIs
  - Step 3: Inference of probabilities of distress (PoD) for each FI under analysis
  - Step 4: Adjustment of PoDs for risk aversion
  - Step 5: Modeling of PoDs as functions of macroeconomic and financial variables
  - Step 6: Modeling of the system’s portfolio multivariate density (PMD)
  - Step 7: Simulation of systemic losses and contribution of individual FIs to systemic risk
  - Step 8: Estimation of financial stability measures and spillovers

- Scenarios and mapping:
  - Baseline and adverse scenarios were considered; macroeconomic scenarios are mapped to macroeconomic and financial variables used to forecast PoDs. The historical distribution of major macroeconomic variables in the scenario (output gap for most PoD explanatory variables) is matched with those of PoD explanatory variables.
- PoD inference choices and rationale:
  - Supervisory data were not available, so market-based information was used.
  - Alternative market-based PoD estimators include the structural approach, CDS-derived PoDs (CDS-PoDs), and out-of-the-money (OOM) option prices.
  - Structural and OOM approaches presented parameterization and data-period difficulties, especially given extreme volatility, mergers/acquisitions, policy interventions, and lack of a full economic cycle in available OOM data.
  - CDS-PoDs were chosen despite limitations (liquidity-driven exaggeration, difficulty disentangling liquidity vs solvency risk) because:
    - CDS spreads reflect market perceptions of distress and frequently anticipate rating changes.
    - CDS-PoDs can be adjusted for risk aversion to approximate real-world PoDs.
    - Once adjusted for risk aversion, CDS-PoDs achieved consistency of SMFST results with historical losses.

- Adjustment for risk aversion:
  - PoDs derived from market-based information are risk neutral and thus were corrected for risk aversion before systemic losses were estimated following Espinoza and Segoviano (2010). See Appendix V for technical details.
  - Figure 15 compares the mean of risk neutral PoD and the mean of adjusted PoD for the system; attempt to isolate fundamental risk from market sentiment is acknowledged as difficult.

- Modeling PoDs and heterogeneity:
  - Factor models were run separately for PoDs of each FI to analyze heterogeneity from portfolio composition, risk profiles, and business models.
  - A summary of statistically significant variables explaining PoD for each FI is presented in Table 9 with estimated coefficients’ sign.
  - Key common sensitivities across FIs (factor model findings):
    - PoDs of all FIs are highly sensitive to:
      - macroeconomic conditions (unemployment and house prices);
      - banking sector’s loan activities (credit and C&I loan index);
      - profitability and funding conditions (Libor spread);
      - risk measures in markets (VIX).
    - Activity in securitization markets, especially in Mortgage Backed Securities (MBS) markets, is significant for some FIs.
    - Systemic risk, as measured by the interlinkage index, is a statistically significant explanatory factor.

- PMD and dependence structure:
  - PoDs are used to model the portfolio multivariate density (PMD) that recovers the joint statistical distribution of implied asset values of the FIs.
  - PMD is recovered using (i) the CIMDO methodology and (ii) the PoD for each FI as inputs. The CIMDO methodology is a non-parametric cross-entropy framework.
  - The PMD captures linear (correlations) and non-linear distress dependence that change throughout the cycle; dependence increases in periods of distress.
  - The dependence structure is characterized by the CIMDO-copula, which changes over time consistent with empirically observed PoDs.

- Loss simulation and risk measures:
  - One-year potential losses are simulated using the PMD to produce the distribution of systemic potential losses.
  - Expected loss (EL) is measured as 50 percentile VaR of this loss distribution and is estimated in line with the Basel definition as
    1
    N
    ii i
    i
    ELSystemExp xPoD xLGD
    
    
    
    where Exp_i, and LGD_i are exposure and loss given default for FI i, respectively.
  - Unexpected losses (UL) are measured as 99 percentile VaR of this loss distribution.
  - Marginal contributions to systemic risk (MCSR) are built on the expected shortfall (ES) at the 95.0 percent confidence.

- Analytical perspectives enabled by PMD:
  - The PMD allows analysis of:
    - (i) common distress in system FIs,
    - (ii) distress between specific FIs,
    - (iii) distress in the system associated with a specific FI.

### Main Results
- Context and risks:
  - Tests were performed amid considerable uncertainty after the financial crisis. While situation stabilized, loss buffers replenished, and pretax income and financial conditions improved in the last two quarters of 2009, supply of credit remains tight and risks include household deleveraging, rising unemployment, and accelerating corporate and commercial property defaults.
- PoD trends:
  - Macroeconomic scenarios imply PoDs are likely to remain at higher levels than pre-crisis in the near future.
  - Towards end-2009, PoDs implied by actual CDS spreads and other market data improved sharply despite weak macroeconomic data.
  - Forecasted PoDs from 2010 are strongly influenced by lagged effects of substantial macroeconomic shocks in 2009 and 2010; from 2011 on projected PoDs gradually moderate with improving conditions.
- Losses and tail vulnerabilities:
  - The SMFST shows tail vulnerabilities remain in the U.S. financial system.
  - ELs decline gradually in both scenarios and reach below end-2007 levels by the end of the stress testing horizon.
  - ULs remain above end-2007 levels for the whole exercise horizon, indicating substantial tail risks.
  - Cumulative EL and UL would be substantial.

### Key statistics and reported tables (as presented)
- Figure 16: Probability of Distress: Minimum, Mean, and Maximum (Baseline and Adverse charts; Sources: Bloomberg and IMF staff estimates.)
- Table 10. Systemic Expected Losses
  - Bn   US D%  Asse ts%  G DPBn  US D%  Asse ts%  G DP
  - 2007
    280.20.2280.20.2
  - 2008
    1251.00.91251.00.9
  - 2009
    820.60.6820.60.6
  - 2010
    750.60.5770.60.5
  - 2011
    340.30.2560.40.4
  - 2012
    200.20.1240.20.1
  - 2013
    210.20.1200.20.1
  - Adve rse  Baseline
  - Sources: Bloomberg, SNL, and IMF staff estimates.

- Table 11. Systemic Extreme Losses
  - Bn USD% Asse ts    % GDPBn USD% Asse ts    % GDP
  - 2007
    1821.51.31821.51.3
  - 2008
    4273.33.04273.33.0
  - 2009
    3302.52.33302.52.3
  - 2010
    3262.52.23312.62.2
  - 2011
    2241.71.52802.21.8
  - 2012
    1911.51.22041.61.3
  - 2013
    1921.51.11921.51.1
  - Memo item
    2009 total equity
    1,0207.97.2
  - VaR 99%VaR 99%
  - Ba se l i n eAd ve r se
  - Sources: Bloomberg, SNL, and IMF staff estimates.

*Source: Bloomberg and IMF staff estimates.*

### 85. Contribution from GSEs to systemic unexpected losses (UL) appears to be

### _cr10244 - 85. Contribution from GSEs to systemic unexpected losses (UL) appears to be

### Contribution of GSEs to systemic UL
- GSEs’ share of total assets in the system is about 13 percent.
- GSE’s share of systemic extreme losses (99 percent VaR) is about 20 percent in 2008 and 2009 (not shown), indicating sizeable distress dependence between GSEs and the system.
- Distress in GSEs appears to cause considerable distress in the system owing to their interconnectedness, despite various policy measures to support them directly.

### Buffers to absorb losses (loan loss reserves and capital)
- From a risk-based perspective:
  - Loan loss reserves represent the buffer to absorb expected losses.
  - Capital represents the buffer available to absorb unexpected losses.
- The largest cumulative losses from 2007 to 2009 were experienced by Citigroup, AIG, Fannie Mae, and Freddie Mac.
- Table 12 (as presented in source):
  - Loan loss reserves
    - 1 520.421170.911621.25
  - Pretax income
    - 2 950.78-225-1.75-57-0.44
  - Total risk-based capital
    - 3 8056.609147.131,0588.16
  - Total equity
    - 8036.588496.621,0207.86
  - Notes shown in source:
    - Source: SNL Financial and staff calculations
    - 2 AIG,FRE,FNM negative pretax income
    - 3 2007-2008 GS, MS & 2007-2009 AIG, FNM, FRE substitute total equity for risk-based capital
    - 1 2009 AIG estimated from Q3 YTD provision expense; 2007-2008 data unavailable for GS, MS - not included in numerator, but included in denominator
    - 200920082007

### Capital gap: definition and evolution
- Capital gap measure (as shown in source):
  - assetstotalSystemic
    ULs
    assetstotalSystemic
    equitytotalSystemic
    CapitalGap
- Capital gap trends (Table 13 values as presented):
  - 2007
    - 6205.094.416205.094.41
  - 2008
    - 4223.292.924223.292.92
  - 2009
    - 6895.314.846895.314.84
  - 2010
    - 6935.354.706885.314.66
  - 2011
    - 7956.135.187405.704.81
  - 2012
    - 8296.395.178166.295.09
  - 2013
    - 8276.384.958276.384.95
- In line with projected UL trends, capital gap continues to improve in both scenarios throughout the exercise horizon, exceeding the 2007 levels already in end-2009 (Table 13).
- Sources cited for Table 13 in the source: Bloomberg, SNL, and IMF staff estimates.
- Note from source: Although alternative measures of capital are presented in Table 11; only total equity and common equity are available for all the institutions considered in this analysis. End-2009 total equity and assets data are used for 2010 on as well.

### Limits and caveats on buffer assumptions
- It is difficult to formulate reasonable assumptions about loss buffers given the amount of uncertainty in the current environment.
- Assuming that current capital levels (in percent of total asset) remain constant through the exercise horizon might look unrealistic.
- The approach used allows a heuristic analysis of the capacity of existing capital to withstand extreme losses while avoiding contamination of estimates with erroneous forecasts of capital buffers.

### MCSR (Marginal Contribution to Systemic Risk) and interconnectedness
- MCSR measured using ES (Expected Shortfall) built on simulated loss distribution accounting for tail distress dependence among the 14 FIs in the system.
- MCSR for a FI i is defined as the difference between the ES with all 14 institutions and the ES with 13 institutions excluding the FI i.
- Findings:
  - Interconnectedness contributes noticeably to a FI’s MCSR in addition to its mere size.
  - The correlation between financial institutions’ asset size and their MCSR:
    - decreased to 0.6 in 2008
    - increased to 0.8 in 2009
  - Relationship between asset size and MCSR decreased considerably in 2008: for some institutions MCSR was larger than relative asset size; for others MCSR was smaller than relative asset size.
  - Changes in MCSR reflect changes in PoDs; in 2008 some institutions’ PoDs increased more than proportionally, increasing their negative spillover effects and thus their MCSR relative to asset size.

### Spillovers and global interconnectedness
- Sample of large U.S., European, and Asian banks analyzed to assess spillover risks from the U.S. financial system to other financial systems (sample includes Citigroup, Bank of America, J.P Morgan, Wachovia, Merrill Lynch, Morgan Stanley, Goldman Sachs, HSBC, RBS, UBS, Deutsche Bank, IB of Korea, ANZ, Mitsubishi UFJ, and Bank of China; estimates performed for data from January 2005 up to December 2009).
- Key results:
  - Vulnerabilities in the global financial system eased from the highest levels observed by end-Q1 2009, but tail systemic risk (global banking stability index) remains at similar levels to those observed during August 2008 for global financial systems.
  - On average, PoDs of U.S. banks remain higher than European and Asian banks.
  - Tight inter-linkages persist between U.S. and European banks; by June 2010, if all the US (European) banks in the sample were to fall into distress, there would be a 40 percent chance that this distress would spill over to European (US) banks.
  - Probability of cascade effects in the global system provoked by distress in U.S. banks remains significantly high: distress of Citigroup, Bank of America, or J.P Morgan would raise the chances of distress of other global banks above 90 percent since 2007.
  - Some institutions that could provoke large systemic risks appear vulnerable to negative developments in other global FIs and emerging market sovereigns. Citigroup’s PoD conditional on distress among major emerging market sovereigns remains at a high level.
  - Spillovers between banks and corporates have been increasing; banks’ distress given corporate distress is higher than corporates’ distress conditional on banks’ distress, reflecting a vulnerable banking system and rising spillovers into the real economy.
  - Global tail systemic risk has eased from early-2009 peaks but remains significantly higher than pre-crisis levels; close interlinkages remain between U.S. and European FIs; probability of cascade effects from U.S. bank distress remains high; interlinkages between U.S. financial and non-financial sectors are increasing.

### Macroprudential stress test conclusions (summary from source)
- Vulnerabilities remain in the U.S. financial system.
- Losses would increase in 2010 and 2011.
- Under the baseline scenario:
  - ELs would have an appreciable increase in 2010; however, to levels below ELs in 2008.
  - ULs would increase to similar levels than ULs in 2008.
- Under the adverse scenario:
  - ELs and ULs would remain at higher levels than the baseline scenario for a prolonged period of time.

*Source: SNL Financial and staff calculations; Bloomberg, SNL, and IMF staff estimates.*

### 104. The present value of market-implied expected losses associated with outstanding

### The present value of market-implied expected losses associated with outstanding

### Valuation of market-implied expected losses
- Market-implied expected losses associated with outstanding liabilities are valued as an implicit put option with the default threshold B as strike on the asset value A of each institution.
- Once asset value A, asset volatility σA, default barrier B, time horizon T, and discount rate r are known, the implicit put option PE(t) can be calculated and decomposed into default probability and LGD.
- An alternative relation links the implicit put option to a debt spread s via a logarithmic transformation (as in the source specification).
- The specification maintains analytical tractability and, as noted, does not incorporate skewness, kurtosis, and stochastic volatility unless alternative closed-form models (e.g., Gram–Charlier) are used for robustness.

### Calibration techniques and implied asset estimation
- CCA models are calibrated using balance sheet information and equity market data to infer:
  - Implied asset value A and implied asset volatility σA from observed equity value E and equity volatility σE.
  - Inputs for the traditional Merton framework include equity value E, equity volatility σE, and the distress barrier.
- State-price density (SPD) of implied asset values can be derived from equity option prices via:
  - Estimation of the risk-neutral density (RND) using mixtures of log-normal densities and nonparametric regression.
  - Closed-form Gram–Charlier model (Backus et al. (2004)) is employed for robustness, allowing kurtosis and skewness without requiring market option prices.

### Measuring market-implied contingent liabilities
- Market-implied contingent liabilities combine the implicit put option from equity and information from CDS markets:
  - If guarantees depress equity values, CDS spreads should capture only expected loss retained by unsecured senior creditors after accounting for implicit guarantees.
  - The scope of government guarantee is defined as the difference between the total expected loss (equity-derived put PE) and the residual default risk on unsecured senior debt (CDS-implied put).
- An adjustment factor equals the ratio between recovery at face value (RFV) and recovery at market value (RMV); in practice this factor is very close to unity for most cases, with a few cases where the factor is within a 10 percent range (0.9 to 1.1).
- Moody’s KMV provides “adjusted liabilities” (short-term debt plus one-half of long-term debt) used to derive the default barrier; MKMV defines this barrier equal to total short-term debt plus one-half of long-term debt.
- CDS-based implied put values can determine the fraction α of total potential loss due to default (PE) covered by implicit guarantees via:
  - α(t) = 1 − (PCDS(t) / PE(t)), where PCDS(t) is the CDS-implied put option value.
- Using daily data, this framework measures time patterns of government contingent liabilities and retained sector risk.

### Caveats, distortions, and tail risks
- Sources of divergence between equity-implied and CDS-implied put values (affecting α) include:
  - Modeling choices and breakdowns in efficient asset pricing during illiquidity.
  - Capital structure changes from crisis interventions (e.g., equity dilution from capital injections).
  - Recovery-at-face-value assumptions in CDS pricing leading to disproportionate spread increases amid rising default risk.
  - Different risk horizons and basis risk between CDS and bond spreads; basis is adjusted over the same maturity of option prices when below-par bonds push up implied CDS recovery.
- Empirical considerations:
  - The option-pricing model may generate biased estimators of expected losses, but this bias tends to wash out for α because both equity- and CDS-implied put values are derived from comparable valuation methods.
  - During distress, alternative option-pricing methods generate expected losses similar to the Merton model; differences emerge as distress abates, making the current contingent-liability specification conservative during the credit crisis.
  - Equity price declines may reflect flight-to-quality motives beyond solvency signals; empirical evidence for the sample shows high cointegration and weaker negative dependence between equity prices and CDS spreads during stress periods, implying consistent co-movement but lower sensitivity of CDS spreads to default-risk changes over time.
- Tail risk treatment:
  - Extreme events during the credit crisis may elude conventional valuation models; alternative option-pricing models and approaches that account for higher moments and rare events are explored to acknowledge unpredictable outcomes.

### Systemic Contingent Claims Analysis (Systemic CCA) methodology
- Objective: Measure joint (systemic) risk of the financial sector by extending CCA from individual firms to a multivariate portfolio of risk-adjusted balance sheets.
- Key elements:
  - Construct multivariate density of market-implied contingent liabilities by combining marginal distributions with a dependence structure (correlation/dependence).
  - Marginal distributions are modeled within the domain of Generalized Extreme Value distributions to identify asymptotic tail behavior and quantify common extreme shocks.
  - Dependence function is estimated iteratively on a unit simplex to optimize coincidence of multiple cross-classified random variables.
- Outputs and uses:
  - Quantify institution-level contributions to systemic risk at different statistical confidence levels.
  - Estimate how systemic risk affects government market-implied contingent liabilities and how policy measures change size and allocation of systemic risk over time.
  - Produce daily point estimates of systemic risk from a time-varying multivariate distribution—argued to be more comprehensive than conditional metrics such as CoVaR and MES.
- Attribution and decomposition:
  - Contributions to joint tail risk are derived as partial derivatives of the multivariate density with respect to changes in relative weights of individual marginal distributions at specified percentiles.
  - Total expected shortfall can be written as a linear combination of individual expected shortfalls, with relative weights given by second-order cross partial derivatives of the inverse joint probability density.

### Applications: forecasting, stress testing, and pricing guarantees
- Scenario analysis and stress testing:
  - Calibrated CCA models support bootstrap procedures simulating counterfactual changes in input variables to assess effects of capital injections (e.g., higher or lower than TARP).
  - A macro-financial stress testing framework provides quarterly estimates of sample institutions from 2010 Q1 to 2014 Q4 for all 28 “surviving” sample banks and insurance companies beyond the historical sample end point (January 2010).
  - Macro-financial linkages are derived from historical sensitivity of monthly market-implied expected losses to selected macro variables using a multivariate dynamic factor model over the entire sample period.
- Pricing a systemic risk surcharge:
  - The fair-value (in basis points) of a risk-based surcharge that compensates for average market-implied contingent liabilities is given by a logarithmic aggregation formula over institutions, involving aggregate default barrier B, risk-free rate r, time horizon T, and the multivariate density function parameters (location μ, scale σ, shape ξ) of individual market-implied contingent liabilities as a time-varying fraction α of expected losses (equity put option PE).
  - The formula aggregates across n institutions and links the VaR percentile and multivariate distribution to a fair-value surcharge.

*Source: IMF staff discussion in the supplied content unit.*

### Box 1. Systemic Contingent Claims Analysis: Calculating Systemic Risk from

### Box 1. Systemic Contingent Claims Analysis: Calculating Systemic Risk from Contingent Liabilities and Expected Losses

### Methodology and Model Specification
- Individual implicit put options (individual estimates of expected losses or individual implicit put options times the alpha-values as individual estimates of market-implied contingent liabilities) are represented as a random vector X,ij of independent and identically distributed (i.i.d.) observations.
- Univariate marginal asymptotic tail behavior is specified by the generalized extreme value (GEV) limiting law:
  - Y_j(x) = F_j^{-1}((1 + xi_j (x - mu_j)/sigma_j)^{-1/xi_j}) with (xi_j, mu_j, sigma_j) parameters and condition (xi_j mu_j sigma_j + ...) as in source text.
- Multivariate dependence structure is defined by A(omega_1, ... , omega_p), derived non-parametrically by expanding Pickands (1981) bivariate logistic method to the multivariate case and adjusting margins per Hall and Tajvidi (2000).
- The multivariate dependence estimator is given by the nonparametric expression:
  - Â_n(omega) = (1/n) sum_{i=1}^n min(1, max( ..., 1 - sum_j y_{ij} ... )) as specified in the source text, subject to omega in the (p-1)-dimensional unit simplex.
- Degree of coincidence of multiple series is established by a Chi-like statistic SR_kp based on the simplex-constrained optimization.
- Maximum likelihood estimation is used to estimate marginal distributions and the dependence structure recursively or on a rolling window, yielding a point estimate of the complete multivariate density Ĝ_{mu,sigma,xi}(.,.,.) at quantile q = 1 - a:
  - ln Ĝ_{mu,sigma,xi}(x, a) = sum_j ln Ĝ_{a, p t j j j} (...) as in source text.

### Expected Shortfall (ES) and Aggregation
- Expected shortfall (conditional Value-at-Risk) is defined as the probability-weighted residual risk beyond threshold probability a (example: a = 0.05 for 95th percentile).
- Average daily ES for a total sample of p institutions:
  - ES_{a,t,p,tau} = E[ ... ] (expression preserved as in source text).
- Threshold quantile definition preserved as in source text:
  - VaR_{q,t,p,t} = sup{ Pr( ... ) > 0.05 } with the mu, sigma, xi arguments as specified.

### Data, Sample, and Inputs
- Sample: 36 institutions including 17 SCAP BHCs, 8 other banks (processing and consumer finance), major broker-dealers, GSEs (Fannie Mae and Freddie Mac until conservatorship on September 8, 2008), AIG, and 8 other insurance groups.
- Estimation period for the primary analysis: daily data January 3, 2007 to January 29, 2009.
- Key inputs: daily market capitalization (Bloomberg), default barrier (Moody’s KMV CreditEdge based on quarterly financial accounts), risk-free rate, one-year time horizon, and CDS spreads (MarkIt).
- Outputs: market-implied expected losses (implicit put option values over a one-year horizon) and market-implied contingent liabilities (alpha * implicit put options).

### Empirical Findings: Univariate and Group Results
- Example dynamics for a bank:
  - alpha(t) near zero up to October 2007, then increased to between 50 and 68 percent.
  - Market-implied expected loss E_P(t) increased to US$150 to 220 billion between October 2008 and August 2009.
  - Market-implied contingent liability E_t P_t alpha_t ranged from US$ 50 to 150 billion (thick red line in source chart).
- Pre-2008 alpha-values for corporates and most financials are near zero (Ford and GE cited as examples).
- Group medians (November 2008 to March 2009):
  - Top 4 banks and investment banks: median market-implied expected loss in the US$ 75 to 100 billion range.
  - Regional, processing and consumer banks: median market-implied expected losses US$3 to 6 billion.
  - Insurance companies: median market-implied expected losses US$ 8 to 15 billion.
  - GSEs: median market-implied expected losses reach US$ 150 billion in mid-2008 (up to conservatorship).
  - Top 4 banks and investment banks: median market-implied contingent liabilities in the US$ 30 to 60 billion range (Nov 2008–Mar 2009).

### Systemic Aggregation, Dependence, and Diversification Effects
- Simple summation of individual market-implied expected losses and contingent liabilities (implying correlation = 1) overstates joint risk by ignoring dependence structure.
- With dependence structure included, the average of the multivariate distribution (50th percentile solid line) is much lower due to a “diversification effect.”
- Expected joint market-implied contingent liabilities (solid line) peaked at about US$140 billion at the end of March 2009, averaging US$74 billion over the sample period.
- Including GSEs in the sample produces a higher 50th percentile estimate (dashed second 50th percentile line).

### Tail Risk and Extreme Outcomes
- After Lehman collapse, extreme tail risk increased sharply as measured by the 95th percentile market-implied expected shortfall.
- Interpretation examples:
  - In November 2008 there was a five percent chance of losses being US$3 trillion over a one-year horizon in the group of 36 financial institutions before SCAP stress test results reduced tail risk to around US$200 billion.
  - Tail risks briefly flared in November following prominent bankruptcies but stabilized to under US$100 billion as of end-January 2010.
- Joint tail risk of market-implied contingent liabilities:
  - Exceeded US$1 trillion in April 2008.
  - Almost reached US$3 trillion in October 2008.
  - April 2008 spike largely reflects market expectations of government support after the Bear Stearns bailout.

### Summary Statistics (Table 14)
- For April 1, 2007 - Jan. 29, 2010:
  - 50th percentile: 75
  - 95th percentile: 144
  - ES (at 95%): 336
- Pre-Crisis: July 1, 2007 - Sept. 15, 2008:
  - 50th percentile: 61
  - 95th percentile: 922
  - ES (at 95%): 46
- Crisis Period 1: Sept. 15 - Dec. 31, 2008:
  - 50th percentile: 48
  - 95th percentile: 315
  - ES (at 95%): 932
- Crisis Period 2: Jan. 1 - May 8, 2009:
  - 50th percentile: 12
  - 95th percentile: 1170
  - ES (at 95%): 290
- Crisis Period 3: May 11, 2009 - Dec. 31, 2009:
  - 50th percentile: 86
  - 95th percentile: 145
  - ES (at 95%): 260
(Note: values presented exactly as in source table; units are US dollar billions where indicated.)

### Early Warning Indicator and Group Contributions
- Systemic CCA can serve as an early warning indicator: periods when average risk (50th percentile) and extreme tail risk (95th percentile) are both high indicate higher probability of systemic risk.
- GSEs were large contributors to systemic risk pre-conservatorship; after GSE conservatorship and after Lehman collapse, BHCs needing additional capital per SCAP contributed far more to systemic tail risk.
- Table 15 summary (Average individual contribution to systemic risk in percent, average ES at 95th percentile; units and formatting preserved as in source):
  - Banks w/ SCAP-identified capital need: Total April 1, 2007-Jan. 29, 2010 = 26.4 (and period splits: Pre-Crisis 102.0; Crisis Period 1 20.7; Crisis Period 2 27.4; Crisis Period 3 30.7; 39.1)
  - Banks w/o SCAP-identified capital need: Total = 7.5 (period splits preserved in source)
  - other: Total = 8.8
  - failed: Total = 14.7
  - Government agencies (GSEs): Total = 23.9 (Pre-Crisis = 38.2; Pre-crisis share 44.5 percent noted in text)
  - Insurance companies: Total = 19.8
  - Aggregate shares sum to 100.0 across periods as reported.

### Scenario Analysis: TARP Counterfactuals and Policy Impact
- Counterfactual framework: calibrate Systemic CCA to measure impact of alternative recapitalization configurations (public capital injections) on systemic risk and government market-implied contingent liabilities.
- Capital injection counterfactual procedure:
  - Estimate event-day sensitivity (three-day event window around announcement) of market capitalization, implied asset values, and asset volatility to government interventions.
  - Bootstrap simulated counterfactual asset paths controlling for joint asymptotic tail dependence.
  - Focused counterfactuals: announcements on October 14 and November 23, 2008 to three major TARP recipients (Capital Purchase Program, Targeted Investment Program, Systemically Significant Failing Institutions Program).
- Simulation results:
  - Capital injections lowered individual market-implied contingent liabilities and reduced systemic tail risk.
  - Supporting the three largest TARP recipients reduced their individual contributions to systemic risk, with benefits slightly outweighing cost of intervention.
  - Doubling the original amount of capital injected would have had little additional effect over time.
  - Without capital support (policy inaction), average market-implied contingent liabilities from the banking sector would have increased by more than 50 percent in 2009 (95 percent VaR) and tail risk and average risk would have escalated substantially (Table 16).

### Scenario Results (Table 16)
- 95th percentile Average Value-at-Risk estimate of market-implied contingent liabilities (in billions of U.S. dollars):
  - April 1, 2007 - Jan. 29, 2010:
    - no capital injections: 238
    - actual: 214
    - 2x capital injections: 197
  - Pre-Crisis: July 1, 2007 - Sept. 15, 2008:
    - no capital injections: 89
    - actual: 91
    - 2x capital injections: 91
  - Crisis Period 1: Sept. 15 - Dec. 31, 2008:
    - no capital injections: 484
    - actual: 479
    - 2x capital injections: 469
  - Crisis Period 2: Jan. 1 - May 8, 2009:
    - no capital injections: 414
    - actual: 359
    - 2x capital injections: 353
  - Crisis Period 3: May 11, 2009 - Dec. 31, 2009:
    - no capital injections: 356
    - actual: 227
    - 2x capital injections: 226

### Stress Testing and Extensions
- Systemic CCA can be linked to a dynamic factor model to perform stress testing by modeling how macroeconomic conditions influence monthly implicit put option changes.
- This linkage allows projecting financial sector performance conditional on macroeconomic paths; results align with findings in Section III of the source.

*Source: IMF staff estimate.*

### 129. In this section, the macro-financial linkages affecting financial sector

### 129. In this section, the macro-financial linkages affecting financial sector

### Methodology: macro-financial linkage and Systemic CCA framework
- Monthly implicit put option values for 36 sample banks and insurance companies between 01/03/2007–01/29/2010 (93 observations) are used as the observable inputs.
- A multivariate dynamic factor model with a vector autoregressive structure for unobserved factors is estimated; macro variables are included as exogenous covariates in both the latent-factor and observable-variable equations.
- Macro variables used: nominal and real GDP growth, real consumption, output gap, unemployment rate, housing prices, ROA in the banking sector, and the three-month-LIBOR-treasury rate spread (“TED spread”).
- The model estimation uses De Jong (1988) for initial Kalman filter values when stationary and the De Jong (1991) diffuse Kalman filter when non-stationary.
- Historical joint sensitivity of monthly implicit put option values is determined at a statistical significance threshold of 10 percent.
- Baseline and adverse scenario extrapolations of implicit put option values are produced from the dynamic factor model; the multivariate density of expected losses and contingent liabilities is estimated over a rolling window of 20 observations, generating quarterly projections through end-2014 according to the Systemic CCA model.
- Sample caveats: several structural changes (mergers, acquisitions, government support) occurred during the credit crisis; 4 out of 36 institutions experienced significant re-organization (Bank of America-Merrill Lynch, J.P. Morgan, Wells Fargo, and Washington Mutual). Calibration at the 10 percent threshold was judged not to be influenced materially by these changes.

### Stress test results — Market-Implied Contingent Liabilities (summary)
- Baseline scenario projections:
  - Median (50th percentile) contingent liabilities projected at around US$31 billion on average.
  - 90th percentile indicates a 10 percent chance of this amount doubling to over US$60 billion.
  - Rising estimation uncertainty until 2014; tail-risk uncertainty considerable at end-2012 (diminishes slowly from 2013 onwards).
- Adverse scenario projections:
  - Median (50th percentile) contingent liabilities projected at around US$41 billion.
  - Great deal of uncertainty in 2011 (90th percentile over US$120 billion and 95th percentile at US$240 billion).
  - Scenario shows improvement by end-2012 and into 2013.

### Stress test results — Market-Implied Expected Losses (summary)
- Baseline scenario:
  - Median projected expected losses rise from US$56 billion to US$81 billion in 2012 Q2, then decline to a little more than one half of that level by 2014.
  - Systemic tail risk (expected shortfall at the 95th percentile) peaks in 2012 Q2 at US$591 billion (almost double the amount estimated for 2010 Q2).
  - Normalization of financial sector to pre-crisis conditions does not occur before 2012 Q3 from a systemic risk perspective.
- Adverse scenario:
  - Median expected losses exceed US$100 billion by 2011 Q1 and then decline only slightly to US$97 billion.
  - Systemic tail risk exceeds the US$3 trillion mark in 2011 Q4, then declines to pre-crisis conditions in 2012 Q3.

### Key numeric results (Table 17: Systemic CCA averages, in billion US dollars; Forecasting Period, 2010 Q1–2014 Q4)
- Baseline Scenario:
  - Market-Implied Contingent Liabilities: 50th percentile 31; VaR (95%) 92; ES (95%) 180
  - Market-Implied Expected Losses: 50th percentile 75; VaR (95%) 219; ES (95%) 429
- Adverse Scenario:
  - Market-Implied Contingent Liabilities: 50th percentile 41; VaR (95%) 130; ES (95%) 382
  - Market-Implied Expected Losses: 50th percentile 97; VaR (95%) 308; ES (95%) 910

### Historical and tail-risk observations (from joint historical analysis)
- Analysis based on daily data for 36 financial firms from 01/01/2007 to 01/31/2009 suggests:
  - Expected joint market-implied contingent liabilities peaked at about US$140 billion at end-March 2009.
  - Average contingent liabilities over the sample period were US$74 billion.
- Joint tail-risk spikes:
  - Spikes in April 2008 and October 2008 indicate high government exposure; 95th percentile expected shortfall of market-implied contingent liabilities exceeded US$1 trillion and almost reached US$3 trillion in those months, respectively.
  - Housing GSEs were large contributors to systemic risk until conservatorship; after Lehman collapse, certain BHCs identified by SCAP contributed far more to systemic tail risk.
- Government capital injections:
  - Simulations indicate capital injections lowered individual market-implied contingent liabilities and systemic tail risk.
  - Capital support to the largest three TARP recipients significantly reduced systemic risk from joint market-implied contingent liabilities; doubling original injections would have had little additional effect over time.
  - In absence of capital injections, tail risk and average risk would have escalated substantially.

### Comparative and sample-adjusted worst-case loss estimates
- Systemic CCA framework captures market expectations of both expected losses and contingent liabilities; Section III covers only expected losses retained in the financial sector.
- After adjustment for different sample sizes (36 institutions with US$14.9 trillion on-balance liabilities vs 12 institutions with US$7.4 trillion), the Systemic CCA framework generates worst-case market-implied loss estimates of more than US$350 billion.
  - Adjustment note: multiply the 95 percent ES from Section III by 0.39 for consistent comparison, resulting in losses of about US$167 billion under the baseline and about $354 billion under the adverse scenario.

### Figures and sampling details
- Projections shown for sample period 2010 Q1–2014 Q4 (20 observations) of quarterly put option values forecasted from monthly put option values between 01/03/2007–01/29/2010 (93 observations).
- Figures present joint expected losses (median and Expected Shortfall at 95th percentile with confidence bands) and joint contingent liabilities (median and percentile ranges).

*Source: IMF staff estimates.*

### 3.3 percent in the Adverse). Short-term market spreads react slightly more than under the

### _cr10244 - 3.3 percent in the Adverse). Short-term market spreads react slightly more than under the

### Scenario outcomes and key projections
- Short-term market spreads react slightly more than under the Adverse in 2010, but return faster to the baseline in the outer years, allowing banks to earn higher profits over the forecast horizon.
- Banks’ assumed difficulty in rolling over maturing debt leads to higher losses on commercial real estate loans, which peak at 5.1 percent at end-2011.

### Macroeconomic assumptions (Table 18 — selected series and numeric sequences preserved exactly)
- Baseline scenario (percent change, unless otherwise noted):
  - Real GDP: 3.1 2.6 2.4 2.5 2.4
  - Real personal consumption expenditures: 2.4 2.1 2.0 2.0 2.0
  - Nominal GDP: 3.9 4.0 4.2 4.4 4.3
  - Output gap (percent): -2.0 -1.0 -0.6 -0.3 -0.1
  - Unemployment rate (percent): 9.8 8.9 7.0 5.8 5.5
  - Case-Shiller 10-city house prices: 2.1 2.0 2.9 2.5 1.5
  - Spread of 3-month LIBOR to 3-month T-Bill: 0.2 0.4 0.4 0.4 0.4
  - Return on assets (annualized; percent): 1.7 1.8 1.8 1.8 1.9
- Adverse scenario (series preserved exactly):
  - Real GDP: 2.3 -0.8 0.8 -1.7 2.6 0.2 2.6 0.1 2.2 -0.2
  - Real personal consumption expenditures: 1.9 -0.6 0.6 -1.6 1.6 -0.4 1.5 -0.5 1.3 -0.6
  - Nominal GDP: 3.8 -0.2 3.4 -0.7 4.1 -0.2 4.6 0.2 4.5 0.2
  - Output gap (percent): -3.0 -1.0 -3.3 -2.3 -2.1 -1.5 -1.1 -0.8 -0.6 -0.4
  - Unemployment rate (percent): 10.0 0.2 9.9 1.0 8.9 1.9 7.7 1.9 6.9 1.5
  - Case-Shiller 10-city house prices: -2.2 -4.3 -2.1 -4.1 2.2 -0.7 2.5 0.0 1.8 0.2
  - Spread of 3-month LIBOR to 3-month T-Bill: 0.3 0.1 0.6 0.3 0.7 0.3 0.6 0.2 0.6 0.2
  - Return on assets (annualized; percent): 1.6 -0.2 1.4 -0.5 1.5 -0.4 1.6 -0.3 1.7 -0.3
- Alternative Funding Risk scenario (series preserved exactly):
  - Real GDP: 2.4 -0.6 0.8 -1.8 1.6 -0.8 2.5 0.0 2.4 0.0
  - Real personal consumption expenditures: 2.3 -0.2 0.2 -1.9 0.0 -2.0 1.3 -0.7 1.7 -0.2
  - Nominal GDP: 3.3 -0.7 1.9 -2.1 3.1 -1.1 4.4 0.0 4.4 0.1
  - Output gap (percent): -3.3 -1.3 -2.6 -1.6 -0.8 -0.3 -0.3 0.0 -0.1 0.0
  - Unemployment rate (percent): 10.6 0.8 9.9 1.0 7.2 0.1 5.8 0.0 5.5 0.0
  - Case-Shiller 10-city house prices: -4.1 -6.1 -2.6 -6.7 3.1 0.3 2.4 0.0 1.5 0.0
  - Spread of 3-month LIBOR to 3-month T-Bill: 0.4 0.1 0.5 0.1 0.5 0.1 0.4 0.0 0.4 0.0
  - Return on assets (annualized; percent): 1.5 -0.3 1.6 -0.3 1.7 -0.2 1.8 -0.1 1.9 -0.1
- Table note preserved: "Shaded numbers denote deviations from baseline."

### Loan loss projection methodology and estimation details
- Charge-off rate modeling:
  - Charge-off rates for different loan types are modeled as dependent on economic and financial variables; levels and log levels were used for explanatory variables to better capture turning points.
  - Cumulative net balances for lending standards ("cumulative lending standards") are used to reflect slower rates of tightening.
  - Output gap (detrended real GDP) used instead of GDP growth to capture deterioration amid negative growth rates.
- Data sources and sample:
  - Loan loss rates and lending standards: Federal Reserve.
  - Macroeconomic and financial data: Haver Analytics.
  - Forecast data where available: WEO.
  - Housing and lending standards modeled separately.
  - Sample: quarterly data from 1991 to 2009.
- Estimation approach:
  - Empirical Bayesian approach to address non-stationarity.
  - Estimation carried out by running 10,000 Markov Chain Monte Carlo simulations using the Gibbs sampler package WinBUGS (Lunn and others, 2000).
  - Convergence obtained within 1,000 burn-in runs.
  - Estimated coefficients reported as statistically significant at 5 percent.
- Two-step approach for real estate charge-offs:
  - Charge-offs = delinquency rate × transition (hazard) rate.
  - Commercial real estate:
    - C_CRE_t = D_CRE_t * T_CRE_t
    - ln(D_CRE_t) = 0.00131*LS_CRE_t – 2.3137*ln(CP_t) + 11.45
    - ln(T_CRE_t) = 0.00117*LS_CRE_t + 1.37
  - Residential real estate:
    - C_RRE_t = D_RRE_t * T_RRE_t
    - ln(D_RRE_t) = 0.00308*LS_RRE_t – 0.0025*HP_t + 0.912
    - ln(T_RRE_t) = 0.00284*LS_RRE_t + 0.12848*UR_{t+4} + 0.997
- Other loan categories:
  - Consumer loans:
    - ln(C_CL_t) = 0.00079*LS_CL_t – 0.0925*GAP_t + 0.705
  - Commercial and industrial loans:
    - ln(C_CI_t) = 0.00341*LS_CI_DT_t – 0.1398*GAP_{t+1} – 0.5752

### Lending standards forecasting
- SLOOS (Senior Loan Officer Opinion Survey) lending standards used as independent variables in charge-off and delinquency models; measures are percentage tightened minus loosened.
- Four VAR models estimated to ground forecasts of lending standards; each VAR included:
  - real GDP growth, CPI inflation, change in corporate bond spreads, change in the Federal Funds Target Rate, oil prices, growth of total bank loans, and one loan category’s lending standards (lagged one quarter).
- Model output (detrended lending standards) suggests loosening in lending standards for all loan types over the forecast horizon; model-based projections were adjusted with judgmental forecasts to align with historical behavior.

### House price modeling and CRE linkage
- S&P/Case-Shiller 10-city house price index specification used:
  - ∆^2 HP_t = -0.7449*∆^2 HP_{t-2} + 0.8184*∆^2 RGDP_t + 0.4605*∆^2 RGDP_{t-1} - 0.0154*∆^2 UR_{t-1}
  - Standard errors and significance annotations preserved as presented: (0.0944)*** (0.2445)*** (0.2392)* (0.0076)**.
  - Definitions preserved: HP = 10-city Case-Shiller index in log levels; RGDP = real GDP; UR = unemployment rate; ∆^2 denotes the second difference.
- Baseline scenario house price outcome:
  - Under the baseline scenario, using output and unemployment forecasts from WEO, house prices rise moderately but are still 24 percent below their 2006:Q2 peak by end-2014.
- Commercial real estate (CRE) price projection:
  - Observation: developments in house prices have generally fed into CRE prices with about a four-quarter lag (MIT Center for Real Estate’s Transactions-Based Index used).
  - CRE prices projected by applying Case-Shiller growth rates to the following year’s CRE series.
  - CRE outcome: commercial real estate prices increase steadily but are 39 percent off their 2007:Q2 peak at the end of the forecast period.
- Correlations (Table 19 preserved):
  - Levels correlations between S&P Case-Shiller house prices (10-city) and CRE at t+1 through t+6: 0.92 0.94 0.95 0.96 0.96 0.95
  - Growth correlations at t+1 through t+6: 0.43 0.33 0.31 0.41 0.49 0.34

### Risk-aversion adjustment, market price of risk, and CDS-derived PoDs
- CDS-PoDs reflect both market expectations of assets’ actual risk and systemic risk aversion (price of risk).
- To estimate actual probabilities of default from risk-neutral PoDs, need to strip out effect of risk aversion using linear pricing and risk-neutral pricing relations (Cochrane, 2001).
- Key relations and notation preserved as given, including the description of risk-neutral probability and the role of m_{t+1}(s), f_t r (risk-free rate), and π̂.
- Espinoza and Segoviano (2010) approach:
  - Use conditional expectation formula for normal distributions to estimate thresholds and conditional expectations (equation structures preserved).
  - Price of risk is related to variance via: var(·) and m_r, and can be calibrated based on the VIX; the threshold can be chosen so that probability the market price of risk exceeds the threshold equals the actual probability of nature.
  - Non-linear equations (1), (2), (3) to be solved jointly (unique solution shown in Espinoza and Segoviano (2010)).

### CIMDO methodology and CIMDO-copula (methodological summary and functional forms preserved)
- CIMDO (Consistent Information Multivariate Density Optimizing) is based on minimum cross-entropy (Kullback, 1959): posterior multivariate distribution p recovered by updating prior q with empirical constraints (FIs’ empirically estimated PoDs).
- Objective functional: C[p,q] = ∫∫ p(x,y) ln( p(x,y) / q(x,y) ) dx dy.
- Consistency-constraint equations impose marginal constraints so posterior marginals match empirically estimated PoDs; constraints written in integral form with indicator functions and distress thresholds preserved.
- Optimization via Lagrange multipliers (λ_1, λ_2 for marginal constraints; μ for additivity) yields posterior multivariate density form (equation structure preserved).
- Intuition: the posterior multivariate distribution (PMD) closest to the prior and consistent with empirically observed PoDs is recovered; CIMDO distributions outperform parametric multivariate densities under the Probability Integral Transformation criterion (see Segoviano (2006) for robustness).
- CIMDO-copula:
  - Copula approach isolates dependence structure by transforming marginals to U(0,1) via Probability Integral Transformation.
  - The CIMDO-derived copula (CIMDO-copula) is a nonlinear function of the optimization multipliers (μ, λ_1, λ_2) and the prior q(x,y); these parameters change as PoDs of individual FIs change.
  - Therefore the CIMDO-copula captures changes in distress dependence automatically as PoDs of individual FIs vary across the economic cycle.

*Source: IMF staff (content unit: _cr10244 - 3.3 percent in the Adverse). Short-term market spreads react slightly more than under the*

### APPENDIX VIII: CONTINGENT CLAIMS ANALYSIS

### APPENDIX VIII: CONTINGENT CLAIMS ANALYSIS

### Contingent claims framework and intuition
- Contingent claims analysis constructs risk-adjusted balance sheets based on three principles:
  - (i) the values of liabilities (equity and debt) are derived from assets;
  - (ii) liabilities have different priority (i.e. senior and junior claims);
  - (iii) assets follow a stochastic process.
- Assets A(t) follow the asset return process dA/A = Aµ dt + Aσ dε, where:
  - Aµ is the drift rate or asset return,
  - Aσ is the standard deviation of the asset return,
  - ε is normally distributed, with zero mean and unit variance.
- Default occurs when assets fall to or below the promised payments Bt. The probability of default is Prob( A(t) ≤ B(t) ).
- “Actual” probability of default (under the physical measure) is given by N(–d2,0) with
  - d2 = [ln(A0/Bt) + (Aµ − 0.5 Aσ^2) t] / (Aσ sqrt(t))
  - (text expresses this as: 2,0 ()Nd− where d is defined above).
- Risk-adjusted (risk-neutral) probability of default substitutes the risk-free rate r for the asset drift µ, yielding a larger risk-adjusted probability of default whenever µ > r.

### Risk-adjusted vs. actual default probabilities
- The asset-return probability distribution used to value contingent claims is the risk-neutral distribution (expected rate r), not the “actual” distribution (expected return µ).
- Implication: risk-neutral (risk-adjusted) probability of default is larger than the actual probability of default for assets with positive risk premium (µ > r).
- The CCA/Merton model uses risk-neutral probabilities for valuation and risk calculations; actual probabilities can be combined with equilibrium models of expected returns to obtain consistent estimates of expected returns on derivatives.

### Technical description of the Systemic CCA methodology
- Individual implicit put options (or implicit put option times alpha-values as contingent liabilities) are represented as an i.i.d. multivariate random vector X = (X1,...,Xpn).
- Extreme-value asymptotic tail behavior of each element Xij is specified via limiting laws of normalized maxima: for constants an > 0 and bn, the sequence converges to a non-degenerate limit distribution GX as n → ∞ (Vandewalle et al., 2004; Stephenson, 2002).
- Univariate marginal distributions converge to generalized extreme value (GEV) or unit exponential forms; each marginal j has parameters:
  - location parameter µj,
  - scale parameter σj > 0,
  - shape parameter ξj.
- Role of the shape parameter ξj:
  - The absolute value of ξj determines tail weight and the speed at which the tail approaches 0.
  - ξ also indicates the number of finite moments (example: if ξ = 2, mean and variance exist; higher moments may be infinite).

### Multivariate dependence specification and estimation
- Rather than a single-parameter copula, the multivariate dependence structure is specified by a function A(·) = A(ω1,...,ωp), derived non-parametrically by expanding the Pickands (1981) bivariate logistic method to the multivariate case and adjusting margins per Hall and Tajvidi (2000).
- Non-parametric estimator A_n(·) is computed via an empirical aggregation (equations A2 and A3), with normalization:
  - Let •̂ = Σ_j Ŷ_ij / n (notation in text) and optimization over the (p-1)-dimensional unit simplex.
- Because the raw estimator A_n(·) may not satisfy theoretical bounds 0 ≤ A(ω) ≤ 1, define a rationally scaled measure:
  - A'_n(ω) = min(1, max( n^-1 A_n(ω), 1 )) (expressed as equation (A5) in the source) so that A'_n is a convex function on [0,1] with A'_n(0)=A'_n(1)=1 and limits under complete dependence and mutual independence.
- Interpretation:
  - A'(ω) equivalent to ω... implies independence of components.
  - A'(ω) equal to max... implies complete dependence (identical marginal distributions).

### Estimation of the multivariate density and Expected Shortfall (ES)
- Margins and dependence structure can be estimated recursively or with a rolling window of length τ and periodic updating.
- The point estimate of the complete multivariate density Ĝ_{ξ,µ,σ,ω} at quantile q = 1 − a over estimation period τ is given by equation (A6) and (A7), where
  - Ĝ(...) = exp{ − Σ_{j=1}^p Ŷ_j,t,n } (notation preserved from source).
- Expected Shortfall (ES) (conditional VaR) is computed as the probability-weighted residual density beyond a pre-specified statistical significance 1 − q = a (e.g., a = 0.05 for 95th percentile threshold).
  - ES defines the average value of the aggregate implicit put option (or contingent liabilities, controlling for alpha-value) on days when it exceeds q.
- Average daily ES for a sample of p institutions, for each day t, is given by:
  - E_a,t^{P} = ... (expressed in equation (A8); source notation preserved: E_{a, t}^{P}PPGaVaR_{t} with arguments µ,σ,ξ).
- VaR threshold quantile is defined as:
  - sup Pr{...} = 0.05 for a = 0.05 and confidence 1 − a = 0.95 (equation (A9) preserved in source notation).

### Scenario analysis steps (estimation of impact of counterfactual policies)
- Step 1: Estimate the bivariate extreme value distribution (EVD) (Pickands, 1990) between pairs of CCA input parameters underlying individual put option values (equity and implied assets; implied assets and asset volatility). This matches both variables conditional on actual joint occurrence over the historical sample.
- Step 2: Let G(.) be the fitted bivariate distribution with margins G1 and G2; define the quantile function for probability p as:
  - x̂_{p} = G^{-1}(p) with margins converging to GEV (notation preserved in source).
- Step 3: Derive a modified bivariate GEV by matching the first moment so theoretical and empirical percentiles of parameter distributions are identical on the day of intervention (controls for intertemporal change in default barrier).
- Step 4: Determine impact of capital injection(s) on CCA input parameters (including default barrier) for an institution over a three-day event window (day before, day of, day after intervention) by calculating sensitivity of equity price and implied asset value to US$1 billion of capital injected.
- Step 5: Derive counterfactual asset process of CCA input parameters using bootstrap/resampling methods (Efron, Tibshirani references in source).
  - Bootstrap the sample mean and associated confidence intervals of the sample mean of all “shocked” CCA input parameters using a rolling estimation window starting 60 days prior to intervention, for each day after intervention.
  - Adjust estimated asset process for dilution effect on day of intervention and condition the asset process on subsequent capital injections (example: Citigroup and Bank of America received US$45 billion and US$45 billion under Capital Purchase Program and TIP).
- Step 6: Re-calculate revised put option value using updated CCA input parameters to derive expected losses and contingent liabilities for each institution.
- Step 7: Re-estimate multivariate density of all put option values using the Systemic CCA framework.

### Key technical properties and notes
- ES is preferred over VaR because ES is coherent (satisfies sub-additivity, convexity, homogeneity), whereas VaR is “incoherent.”
- However, conditioning ES on the most severe outcomes for the entire sample (as in some implementations) can ignore optionality from a wide range of underlying asset values below the ES threshold.
- The Systemic CCA framework allows for:
  - daily updating,
  - multivariate extreme value modeling with time-varying, non-linear dependence,
  - forward-looking market information (equity and CDS) to compute individual contingent liabilities and systemic contingent liabilities.

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*APPENDIX VIII: CONTINGENT CLAIMS ANALYSIS (excerpt).*

### APPENDIX X: USING CCA TO CALCULATE A POSSIBLE SYSTEMIC RISK SURCHARGE

### APPENDIX X: USING CCA TO CALCULATE A POSSIBLE SYSTEMIC RISK SURCHARGE

### Overview
- Discusses debate over systemic risk surcharges, fees, and taxes, and whether charges should be ex-post or ex-ante and whether proceeds should go to special funds or general government revenue.
- Notes CCA (Contingent Claims Analysis) estimates of government contingent liabilities from the financial sector can be used to calculate a “fair value” price of a systemic risk surcharge.
- References systemic-risk-based surcharge approaches, including systemic capital surcharges and risk-budgeting approaches, as discussed in Chapter 2 in IMF (2010).

### Methodology / Formulation
- The fair value (in basis points) of a risk-based surcharge that would compensate for the average contingent liabilities can be written:
  - 
    
    
    
    
    
    
    
    
    
    
    1
    ,,,
    1
    ;
    1
    ln   110, 000
    pT
    p
    rT
    j
    j
    GaP
    T
    T
    Be
  - In the notation:
    - B represents the aggregate default barrier of all p-institutions in the sample.
    - r is the risk-free rate.
    - T is time horizon of the surcharge.
    - 
      
      1
      ,,
      .G is the multivariate density function (with location, scale and shape parameters , , and ) of individual contingent liabilities as a time-varying fraction  of expected losses ,pT P (equity put option).
- Suggests fair value price can be calculated from risk indicators derived from models that quantify magnitude of risk transfer from the financial sector to the sovereign balance sheet, such as the Systemic CCA model.
- Notes it is possible to devise a counter-cyclical surcharge by combining estimates at different percentile levels of statistical confidence (footnote 62).

### Illustrative Estimates (CCA results)
- Using results obtained from the CCA analysis since April 1, 2007 (Section IV):
  - The estimated average annual systemic surcharge for systemically important financial institutions would at least be 49 basis points.
    - This reflects a fair value charge to pay back the government for the implicit and explicit liability guarantees it provided over the period of the crisis.
    - This charge would be on debt liabilities excluding insured deposits.
  - A reasonable average systemic surcharge for systemically important financial institutions would be 39 basis points per year if based on observations between July 2007 and September 2008 before the Lehman crisis.

### Key Statistics — Table 21: United States: Estimated Fair Value Surcharge for Systemic Risk Based on Total Contingent Liabilities from the Financial Sector
- Columns: Systemic Cont. Liabilities - 50th percentile (US$ billion; annual fee (basis points)) | Systemic Contingent Liabilities - 95th percentile (US$ billion; annual fee (basis points))
- Period-level estimates:
  - April 1, 2007 - Jan. 29, 2010
    - 74; 49
    - 214; 142
  - Pre-Crisis: July 1, 2007-Sept. 15, 2008
    - 59; 39
    - 91; 60
  - Crisis Period 1: Sept. 15-Dec. 31, 2008
    - 43; 28
    - 479; 317
  - Crisis Period 2: Jan. 1-May 8, 2009
    - 119; 76
    - 359; 232
  - Crisis Period 3: May 11, 2009-Dec. 31, 2009
    - 88; 58
    - 227; 150
- Source: IMF staff estimates.

### Policy implications and uses
- The CCA-derived fair value surcharge provides a quantified basis for charging systemically important financial institutions for the expected fiscal cost of implicit and explicit government guarantees.
- Surcharges can be structured:
  - As an annual fee in basis points on debt liabilities excluding insured deposits.
  - With counter-cyclical features by using time-variation and percentile-based estimates of contingent liabilities.
- The framework supports risk-based pricing of guarantees and potential design choices between ex-ante vs ex-post charging and allocation of proceeds to funds or general revenue.

*Source: APPENDIX X: USING CCA TO CALCULATE A POSSIBLE SYSTEMIC RISK SURCHARGE (IMF staff estimates).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/scr/2010/_cr10244.pdf_