## 1. General Systemic Risk Measurement Approaches. ## Source details **Canonical URL:** [1. General Systemic Risk Measurement Approaches.](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp1354.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp1354.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp1354.pdf.json) --- ### Systemic risk, policy context, and measurement approaches - Systemic risk definition and MPS objectives: - MPS aims to limit, mitigate or reduce systemic risk, minimizing the incidence and impact of disruptions in the provision of key financial services that can have adverse consequences for the real economy. - MPS is predicated on: (i) assessment of system-wide vulnerabilities and identification of threats from build-up/unwinding of financial imbalances, (ii) shared exposures to macro-financial shocks, and (iii) possible contagion/spillover effects from individual institutions and markets due to direct or indirect connectedness. - Systemic risk: individual or collective financial arrangements—both institutional and market-based—that could lead directly to system-wide distress in the financial sector and/or significantly amplify its consequences with adverse effects on other sectors. - Institutional responses and complementary measures: - More stringent prudential standards (limits on leverage, higher capital requirements). - Broader adoption of contingent capital initiatives (including mandatory debt-to-equity “bail-in” provisions). - “Living wills” and strengthened resolution processes for LCFIs. - Specialized macro-prudential supervisors (examples cited: FSOC, ESRB, FPC). - Two conceptual measurement approaches: - Contribution approach (“Risk Agitation”): - Concept: systemic resilience to individual failure; transmission = "institution-to-institution". - Key features: size, connectedness, complexity, risk-bearing capacity, substitutability. - Policy objective: avoid/mitigate contagion effect; avoid moral hazard. - Participation approach (“Risk Amplification”): - Concept: individual resilience to common shock; transmission = "institution-to-aggregate". - Key features: market risk exposure, asset liquidation, maturity mismatches, debt pressure. - Policy objective: maintain overall functioning of system and maximize survivorship. - Literature and methods: - Institution-level contribution measures include CoVaR, CoRisk, SES, DIP, Granger Causality, SRISK, Joint Probability of Distress. - Network and agent-based models align with participation approach. - Gap identified: few models estimate multivariate firm-by-firm dependence with structural default modeling prior to Systemic CCA. ### Methodological motivation for Systemic CCA - Purpose: - Address lack of forward-looking, market-implied multivariate measures of joint default risk with a structural default model. - Core idea: - View a sample of firms as a portfolio of individual expected losses (firm-specific risk parameters) and model their joint tail risk via a multivariate density of combined expected losses. - Identify endogenous linkages affecting joint expected losses during stress, accounting for time-varying dependence. - Two-step estimation architecture: 1. Estimate each firm’s expected losses (change in capital levels) using an enhanced Contingent Claims Analysis (CCA) that values expected losses as implicit put options. 2. Assume firm-level expected losses follow a Generalized Extreme Value (GEV) distribution and combine them using a non-parametric dependence measure to derive joint expected losses as a multivariate conditional tail expectation (CTE). - Main Systemic CCA features: - Extends risk-adjusted balance sheet CCA to determine systemic risk magnitude from interlinkages based on time-varying likelihood of joint declines of implied asset values below the debt-driven “default barrier.” - Quantifies individual contributions to systemic solvency risk and spillover risks as contingent liabilities. - Treats solvency risk via stochastic capital assessment using option-pricing models and market-implied asset values/volatilities. - Estimates non-linear, non-parametric dependence between firms; linkages are endogenous and dynamic. - Data requirements: accounting on outstanding liabilities, market data on equity and equity option prices, market rates, macro data for satellite stress-test models. - Strengths: integrates market-implied expected losses, endogenizes LGD in a multivariate joint-default specification, quantifies time-varying individual contributions, can price systemic risk charges. - Weaknesses: requires assumptions on option-pricing model specification; complex and resource-intensive. ### Positioning relative to other systemic risk measures - MES/SES and SRISK: - MES/SES generate empirical linear/bivariate dependence measures; SRISK uses a closed-form specification (DCC-GARCH) for correlations but differs from multivariate density estimation. - Systemic CCA contribution: - Provides a multivariate density framework allowing determination of marginal contribution to concurrent changes in severity and dependence across any set of institutions for any confidence level and time. --- ### Step 1 — Calculating expected losses from Contingent Claims Analysis (CCA) - Core CCA/BSM framework: - Equity holders modeled as a call option on firm value; expected losses to creditors valued as an implicit put option with strike equal to present value of promised risky debt payments. - Market identity: A(t) = E(t) + D (risky debt maturing at T). - Default occurs if asset value falls below default barrier (present value of promised debt B). - Conventional expected loss accounting: PD × LGD × EAD. - Risk-adjusted (CCA) balance sheet vs traditional accounting: - Risk-adjusted assets: implied market value A; liabilities: risky debt D (default-free debt minus expected losses) and observable market equity E. - Under CCA, declines in asset value increase expected losses to creditors and lead to less-than-one-to-one declines in market equity. - Valuation of expected losses as an implicit put option: - Put option value depends on A(t), B, r, σ, Φ, and duration T−t (source equations referenced). - Observable inputs: Eσ (equity volatility), Et (equity price), r (risk-free rate), Φ (standard normal CDF), duration T−t. - Asset value and asset volatility can be derived from equity and equity option prices; peer-group scaling/historical methods for non-listed firms. - Option-pricing enhancements (addressing BSM shortcomings): - Gram-Charlier (GC) expansion introduces skewness (γ1) and kurtosis (γ2) corrections. - Jump-diffusion adds Poisson jump intensity λ with log-normal jump size mean φ and volatility ν; put value expressed as infinite series conditional on k jumps. - Stochastic volatility models (Heston), GARCH, copulas, Levy processes, non-parametric methods, lattice/binomial trees also noted. - Empirical note: stochastic volatility yields the most pricing improvement; jumps yield smaller improvements. - Estimation practices and caveats: - Alternative: derive implied asset value from state-price density (SPD) estimated from option prices (Breeden and Litzenberger, Aït-Sahalia and Lo approach). - Sources of uncertainty: distributional assumptions, implied asset/volatility derivation, default barrier assumptions, equity prices reflecting non-fundamental factors. - Convergence: if asset volatility is zero, risk-adjusted balance sheet converges to accounting balance sheet (expected loss → 0). --- ### Step 2 — Estimating joint expected losses from default risk - Overview: - Combine firm-level put option values (expected losses) into system-wide default risk. - Marginals assumed fat-tailed per extreme value theory (EVT); dependence modeled non-parametrically; tail risk measured via VaR/ES. - (i) Marginal distributions: - Observations: X = (EmP1(t), ..., EmPn(t)) estimated over rolling window of τ observations (example: daily sliding window of 120 days). - GEV types: Gumbel (EV0), Fréchet (EV1), negative Weibull (EV2). - Unified GEV PDF/CDF used (equations referenced); jth density f̂_j(x) with parameters μ_j (location), σ_j (scale, >0), ξ_j (shape, ξ_j ≠ 0). - Moments expressed with hp = Γ(1 − p ξ) and other functions as in source (equations preserved). - Estimation: maximum likelihood (ML) over rolling windows; LRS estimator used as initial value; ML fails for 1 ξ ≤ − (no global maximum). - (ii) Dependence structure: - Non-parametric multivariate dependence extends Pickands (1981) logistic method to multivariate with Hall and Tajvidi (2000) margin adjustment. - Dependence function γ(·) convex on [0,1] with γ(0)=γ(1)=1; factor weights ω_j ∈ [0,1] constrained on unit simplex S_m (equation (18)). - γ estimated iteratively over rolling windows (example τ = 120 days) via optimization on S_m. - Contrasts with time-invariant copula approaches; explicitly captures time-varying coincidence across series. - (iii) Joint distribution: - Margins combined with γ to form multivariate extreme value distribution (MGEV) over same estimation period. - Joint CDF and joint density functions specified (equations referenced); joint likelihood maximized over all parameters θ = (μ,σ,ξ). - (iv) Tail risk measure (ES / conditional VaR): - ES_a(X) = E[X | X > VaR_a(X)] with integral representation (equation (22)); ES satisfies coherence axioms H1–H4 (subadditivity, monotonicity, positive homogeneity, translation invariance). - Multivariate ES_{a,τ}(·) computed via integrals over joint distribution G and its quantile G^{-1}_a using γ and MGEV (equation (23)). - Joint VaR condition example: sup{ z : Pr(G^{-1}_a ≤ z) ≥ 0.95 } (equation (24)). - Point estimate of joint potential losses for m firms at time t given explicitly as function of ξ̂, μ̂, σ̂, ω and τ (equation (25)). - (v) Individual contribution to joint expected losses: - Contribution derived from cross-partial derivative of G^{-1} w.r.t γ and marginal severity y_{t,j}. - Joint ES decomposed: ES_{a,τ} = ∑_{j=1}^m ψ_{a,τ}^{(j)} ES_{t,j,a,τ} (equation (26)). - Relative weight ψ_{t,τ}^{(j)} defined via ∑_{s=1}^m ∂^2 G^{-1}_a / (∂γ ∂y_{t,j}) evaluated at percentile; constraint ∑_j ψ_{t,τ}^{(j)} = 1 (equation (27)). - Use cases: identifies marginal impact of each institution on systemic tail losses; evaluates effects of remedial actions. ### Extensions — price-based macroprudential measure and insurance premium - Concept: - Price an insurance premium indemnifying a firm’s marginal contribution to systemic risk using the firm-specific ψ weights and aggregate default barrier B. - Hazard-rate based relations (equations preserved from source): - φ_{t,a,τ}^{(j)} = − ln(1 − B_{t}^{−1} ∑_{·} ψ_{·} ES_{·}) (equation (28) preserved in form). - Survival/hazard relation: φ(t) = 1 − exp(−∫_0^t u(s) ds) (equation (29)). - Insurance monetary cost on short-term liabilities: Insurance_{ST,t,j} = φ_{t}^{(j)} × B_{ST,avg} with the ln and 110,000 scaling factor preserved as shown in the source expression (equation (30) referenced). - Policy considerations: - Premium internalizes own-default externalities and aims to change firm behavior. - Implementation risks: additional burden when capital scarce; institutionalizing systemic importance could exacerbate moral hazard; calibration and long implementation needed; questions about ex ante vs ex post charging and proceeds destination. --- ### Sovereign–bank feedback and contingent liabilities (Box 2 and Systemic CCA sovereign linkage) - Sovereign–bank negative feedback: - Decline in market value of sovereign debt weakens implicit sovereign guarantees, raises banking sector default risk, increases borrowing costs, and raises likelihood of downgrades to financial institutions. - Lender–borrower channel when banks hold large public sector exposures accentuates linkage. - Market-implied contingent liabilities via CCA: - Government guarantee heuristically ≈ E_Pt − CDS_Pt where E_Pt is total expected loss (put value) and CDS_Pt is put implied by CDS spread. - alpha(t) = 1 − CDS_Pt / E_Pt defines share of expected loss covered by implicit/explicit government guarantees. - Caveats: estimation sensitive to modeling choices, market illiquidity, equity dilution, recovery assumptions, risk horizons; basis-risk adjustments referenced. - Sovereign CCA calibration: - Sovereign CDS spread and implied sovereign put option calibrated using term structure tenors 1, 3, 5, 7, and 10 years. - Sovereign asset decomposition: sov_At = R + PV(primary fiscal surplus) − bank_t P_bank α + Other (R = foreign currency reserves; implicit/explicit contingent liabilities to banking sector = bank_t P α). - Feedback channels: - Higher sovereign spreads → α decreases → bank spreads increase → bank put option increases → bank default barrier may increase via higher borrowing cost premium δ. - Integrated market-implied capital assessment: - MCAR (Market-implied capital adequacy ratio) = Market Capitalization / Implied Assets. - EL Ratio = Expected Losses (from CCA) / Market Capitalization. - Aggregate counterparts: Aggregate MCAR; Joint EL Ratio = Joint Expected Losses / Aggregate Market Capitalization (example percentile: 95th). --- ### Sensitivity, market-implied credit spreads, and stress-testing framework (Section V) - Fair value credit spreads and non-linear dynamics: - Higher expected losses → lower implied asset value and higher implied asset volatility → higher equity risk premium → EL ratio rises and MCAR decreases → higher fair value credit spread (equations (37)–(40) referenced). - Market-implied capital shortfall: - Defined as capital required to maintain solvency under stress irrespective of changes in expected losses, net of existing common equity buffer. - Allows reconciliation with prudential solvency standards. - Systemic CCA for stress testing: - Projects firm-specific market-implied expected losses over forecast horizon and combines to define system-wide solvency risk. - Firm-specific CCA-based measures empirically used: - (i) contingent liabilities as share of expected losses potentially transferred to public sector, - (ii) capital shortfall based on expected losses in excess of existing common Tier 1 capital above regulatory minimum, - (iii) capital shortfall based on MCAR generated from market capitalization change relative to asset value. - Linking to macro-financial scenarios: - Satellite models estimate historical sensitivity of expected losses to macro/bank variables (dynamic panel regression). - Structural model adjusts implied asset values by forecasts of net operating income and re-estimates implied asset volatility to derive revised put values. - Multivariate aggregation: combine univariate marginals of forecasted expected losses and dependence to derive joint contingent liabilities and capital shortfall. --- ### Empirical applications and key quantitative findings - U.S. financial system empirical application (IMF FSAP context): - Sample: 33 large commercial banks, investment banks, insurance companies, and special purpose financial institutions. - Daily data: January 1, 2007—end-January 2010; sample period for charts: 01/03/2007-01/29/2010. - Key inputs/parameters: - daily implied asset values from SPD using equity option data (Bloomberg), - default barrier from quarterly financial accounts (Moody’s KMV CreditEdge), - risk-free rate r=3.0 percent, - time horizon T=1 (one-year), - one-year CDS spreads (MarkIt), - default barrier = total short-term debt + one-half of long-term debt (Moody’s KMV convention), - SPD from daily European-style equity call option price data with time-to-maturity three months, - multivariate distribution estimated over rolling window of 60 working days with daily updating. - Empirical diagnostics: - Alpha-value dynamics: median and inter-quartile range across 29 sample firms with significant contingent liabilities show secular increase and decreasing dispersion; sudden declines around April 2008, October 2008, May 2009. - Aggregate Systemic CCA (Figure 6): multivariate density from GEV marginals and non-parametric time-varying dependence; marginal severity and dependence estimated over 60 working days. - U.S. contingent liabilities and systemic risk (Appendix 3 / Figure 6 and Figure 7 findings): - Sum of individual contingent liabilities: - peaks at more than 40 percent of GDP at the end of February 2009, - averages 2.5 percent of GDP over the entire sample period. - Extreme tail risk (95th percentile ES) of expected losses transferred to the government: - exceeded nine percent of GDP in April 2008, - almost reached 20 percent of GDP in October 2008. - Interpretation: markets implied joint contingent liabilities of 20 percent of end-2009 GDP with probability < five percent over a one-year horizon during exceptional systemic distress. - Median aggregate contingent liabilities (controlling for dependence) dropped significantly relative to naïve summation. - Individual contributions and systemic surcharge (Figure 8, Table 6, Table 7): - Group contributions to average ES at 95th percentile (selected entries, percent): - Banks: Pre-Crisis (July 1, 2007—Sept. 14, 2008) = 15.6; Total Period (April 1, 2007—Jan. 29, 2010) = 27.2 - Insurance companies: Pre-Crisis = 13.9; Total Period = 15.0 - Other non-bank financial institutions: Pre-Crisis = 3.0; Total Period = 14.7 - Failed (or bailed-out) financial institutions: Pre-Crisis = 69.5; Total Period = 43.1 - Naïve summation would have overstated failed/rescued institutions’ tail-risk contribution by 43.5 percentage points on average and understated other banks by 19.5 percentage points and insurance by 10.2 percentage points. - Systemic surcharge (insurance premium) results: - Annual surcharge for systemically important institutions about 50 basis points on average. - Pre-crisis (July 1, 2007—Sept. 14, 2008): average surcharge 39 basis points per year; Amount = 59 (In billion U.S. dollars) at 50th percentile. - Crisis (Sept. 15—Dec. 31, 2008): Amount = 432; Annual fee/surcharge = 479 (95th percentile). - Total Period (April 1, 2007—Jan. 29, 2010): Amount = 744; Annual fee/surcharge = 214 (50th percentile); 142 (95th percentile). - At the 95th percentile, surcharge shortly before Lehman would have been 60 basis points and shortly after Lehman 317 basis points. - Controlling for insured deposit share reduces default barrier B by 15 percent on average, lowering estimated surcharge. - U.K. banking sector stress tests (IMF FSAP application): - Sample: five largest commercial banks, largest building society, largest foreign retail bank; daily data January 3, 2005—end-March 2011. - Forecast horizon: 2011–2015; joint capital shortfall estimated with five-year sliding window, monthly updates. - Key scenario quantitative findings (preserved exactly): - Joint capital losses under prolonged slow growth scenario: up to 2 percent of end-2010 GDP (£29 billion). - Translates to average capital shortfall up to 0.4 percent of end-2010 GDP (£6.3 billion). - Baseline to mild/severe double-dip (without debt haircuts, from 2013 onwards): potential losses could be zero. - Severe double-dip with sovereign and bank debt haircuts in 2011: estimated potential losses between 6.4–7.1 percent of end-2010 GDP, or £94–104 billion (depending on satellite model). - Estimated capital shortfall relative to Basel III Tier 1 hurdle rates: between 4.4–5.0 percent of end-2010 GDP (£63–73 billion). - Relative to FSA interim capital regime Tier 1 requirements: between 4.9–5.6 percent of end-2010 GDP. - Earlier reported estimates: 1.6 percent of end-2010 GDP relative to the Basel III Tier 1 hurdle rates (and 1.6–1.8 percent relative to the FSA interim Tier 1 requirements) depending on satellite model. - Severe double-dip recession assigned a two percent probability; joint realization of capital losses beyond the 95th percentile implies probability < 0.1 percent (0.02(1−0.95)=0.001). - Distribution of individual bank contributions to joint capital shortfall (sample period 01/03/2005-03/29/2010; percent of joint capital shortfall, average per time period): - Minimum (Average): 0.6 - 25th percentile (Average): 2.0 - Median (Average): 5.7 - 75th percentile (Average): 17.2 - Maximum (Average): 55.3 - Tail-of-tail results: - Severe adverse scenario could have resulted in average joint capital losses up to 3.4 percent of 2010 GDP (£50 billion); under that scenario estimated capital shortfall could have caused average capital shortfall between 1.3– (text truncated in source) with other figures preserved above. --- ### Appendices and technical estimation notes (selected) - Appendix 1 (CCA definition and formulas): - Asset dynamics under risk-neutral measure: dA(t) = r A(t) dt + A(t) dWt σA (equation (A1.1)). - Distance to Default (DD): d = [ln(At/B) + (r + 1/2 σA^2)(T−t)] / [σA sqrt(T−t)] (A1.6). - Implicit put option and PD×LGD decomposition (A1.16, A1.22). - Dividend “lumpy” treatment and volatility adjustments specified (A1.10–A1.14). - Appendix 2 (SPD estimation via Breeden-Litzenberger): - Arrow-Debreu prices via butterfly spreads; f*(AT) = e^{r(T−t)} ∂^2 C/∂K^2 evaluated at K = AT (equations A2.6–A2.7). - Appendix 3 (LRS estimator for GEV shape parameter): - LRS estimator ξ̂ specified (equation (A3.1)); v_{n,i}^c and constants defined (A3.2–A3.3). - Appendix 4 (Moody’s KMV re-expression under risk neutrality): - KMV EDF adjusted for market price of risk; quadratic in adjusted asset volatility σ_A^* and solution structure provided (A4.3–A4.5). - Appendix 5 (comparison to other systemic risk measures): - CoVaR, MES/SES, DIP, and Systemic CCA compared by inputs, dependence modeling, and interpretability. - Key claims: Systemic CCA yields multivariate density and ES-based CTE, quantifies marginal contributions at any confidence level, uses market-implied balance-sheet structure. *Italic: Excerpted from IMF Working Paper content unit _wp1354 — "1. General Systemic Risk Measurement Approaches."* ### 1. General Systemic Risk Measurement Approaches. ...............................................................6 ### 1. General Systemic Risk Measurement Approaches. ### Major items referenced in the source PDF (document-level inventory) - Section title and page location: - "1. General Systemic Risk Measurement Approaches." — page 6 - Other main sections listed in the document: - "2. Selected Institution-Level Systemic Risk Models." — page 8 - "3. Main Features of the Systemic CCA Model." — page 12 - "4. Traditional Accounting Bank Balance Sheet." — page 17 - "5. Risk-adjusted (CCA) Bank Balance Sheet." — page 18 - "6. United States: Systemic CCA Estimates of Market-Implied Average Individual Contribution to Systemic Risk from Contingent Liabilities" — page 52 - "7. United States: Systemic CCA Estimates of Market-Implied Fair Value Surcharge for Systemic Risk based on Total Contingent Liabilities" — page 54 - "8. United Kingdom: Systemic CCA Estimates of Market-Implied Joint Potential Capital Loss" — page 63 - "9. United Kingdom: Systemic CCA Estimates of Market-Implied Individual Contributions of Sample Banks to Systemic Risk—Market-Implied Joint Capital Loss" — page 64 ### Figures included in the document (by number and title) - Figure 1. The Location of Expected Shortfall (ES) in a Stylized Loss Distribution — page 28 - Figure 2. Valuation Linkages between the Sovereign and Banking Sector — page 34 - Figure 3. Integrated Market-based Capital Assessment Using CCA and Systemic CCA — page 40 - Figure 4. Key Conceptual Differences in Loss Measurements — page 46 - Figure 5. United States: Financial Sector – Time Pattern of the Alpha-Value — page 48 - Figure 6. United States: Systemic CCA Estimates of Market-Implied Total Contingent Liabilities and Multivariate Density of Contingent Liabilities — page 49 - Figure 7. United States: Systemic CCA Estimates of Market-Implied Average Daily Expected Shortfall (ES) — page 51 - Figure 8. United States: Decomposition of Systemic CCA Estimates of Market-Implied Average Daily Expected Shortfall (ES) — page 53 - Figure 9. United Kingdom: Integrated Market-based Capital Assessment of a Single Firm Based on CCA-derived Estimates of Expected Losses — page 56 - Figure 10. United Kingdom: Integrated Market-based Capital Assessment of a Single Firm Based on Systemic CCA-derived Estimates of Joint Expected Losses — page 57 - Figure 11. United Kingdom: Integration of the RAMSI and Systemic CCA Models based on Common Specification of Macro-Financial Linkages — page 60 - Figure 12. United Kingdom: Systemic CCA Estimates of Market-Implied Joint Capital Losses from the U.K. FSAP Update Top-Down Stress Tests — page 62 - Figure A1. Stylized Illustration of the Marginal Rate of Substitution Between Individual and Systemic Risk: Bivariate Kernel Density Function and Contour Plot of Individual Contingent Liabilities and Systemic Risk from Joint Contingent Liabilities (Systemic CCA) — page 92 ### Boxes included in the document (by number and title) - Box 1. Extension of BSM Model Using the Gram-Charlier (GC) Specification or Jump Diffusion — page 21 - Box 2. Interaction and Feedback between the Sovereign and Financial Sector Balance Sheets Using the Systemic CCA Framework — page 37 - Box 3. The Importance of Distributions and Dependence in Stress Testing — page 44 ### Appendices included in the document (by number and title) - Appendix 1. Standard Definition of Contingent Claims Analysis (CCA) — page 78 - Appendix 2. Estimation of the Empirical State Price Density (SPD) — page 84 - Appendix 3. Estimation of the Shape Parameter Using the Linear Combination of Ratios of Spacings (LRS) Method As Initial Value for the Maximum Likelihood (ML) Function — page 86 - Appendix 4. Derivation of the Implied Asset Volatility Using the Moody’s KMV Model — page 87 - Appendix 5. Comparison of Different Systemic Risk Measures vis-à-vis Systemic CCA — page 89 *Source: _wp1354 - 1. General Systemic Risk Measurement Approaches. (PDF), canonical URL provided in source metadata.* ### References ............................................................................................................. ### I. INTRODUCTION ### Systemic risk and macroprudential policy (MPS) - MPS aims to limit, mitigate or reduce systemic risk, minimizing the incidence and impact of disruptions in the provision of key financial services that can have adverse consequences for the real economy (and broader implications for economic growth). - MPS is predicated on: - (i) the assessment of system-wide vulnerabilities and the accurate identification of threats arising from the build-up and unwinding of financial imbalances, - (ii) shared exposures to macro-financial shocks, and - (iii) possible contagion/spillover effects from individual institutions and markets due to direct or indirect connectedness. - Systemic risk: individual or collective financial arrangements—both institutional and market-based—that could either lead directly to system-wide distress in the financial sector and/or significantly amplify its consequences (with adverse effects on other sectors, in particular capital formation in the real economy). (footnotes referenced: 4,5) - Impairment to the flow of financial services includes temporary unavailability of services and sharp increases in the cost of obtaining services; disruptions can originate inside or outside the financial system and generate externalities on economic activity. (footnote 6) ### Policy context and institutional responses - Current policy efforts: shift from microprudential focus on individual institution viability toward a market-wide perspective of supervision and comprehensive assessment of systemic risk. - Complementary measures include: - more stringent prudential standards (limits on leverage, higher capital requirements), - broader adoption of contingent capital initiatives (including mandatory debt-to-equity “bail-in” provisions), - “living wills” and strengthened resolution processes for large complex financial institutions (LCFIs), - establishment of specialized macro-prudential supervisors (e.g., Financial Stability Oversight Council (FSOC) in the United States, European Systemic Risk Board (ESRB), Financial Policy Committee (FPC) in the United Kingdom). (footnote 9) - Successful implementation depends on quality of surveillance/analytical tools, institutional strength of supervisory measures, and effectiveness of policy instruments (including persuasiveness of recommendations). ### Two conceptual measurement approaches to systemic risk (Table 1 summary) - Contribution approach (“Risk Agitation”): - Concept: systemic resilience to individual failure. - Description: a contribution to systemic risk conditional on individual failure due to knock-on effects. - Transmission: "institution-to-institution". - Key features: size, connectedness, complexity, risk-bearing capacity, substitutability. - Policy objective: avoid/mitigate contagion effect (contain systemic impact upon failure), avoid moral hazard. - Participation approach (“Risk Amplification”): - Concept: individual resilience to common shock. - Description: expected loss from systemic event due to common exposure and risk concentration. - Transmission: "institution-to-aggregate". - Key features: market risk exposure, asset liquidation, maturity mismatches, debt pressure. - Policy objective: maintain overall functioning of system and maximize survivorship, preserve collective burden sharing. - Note: policy objectives and indicators are not exclusive; measurement approach customization depends on data availability and financial institution characteristics. ### State of the literature and empirical methods - Prominent institution-level measurement approaches (mainly contribution approach) include: - CoVaR (Adrian and Brunnermeier, 2008), - CoRisk (Chan-Lau, 2010), - Systemic Expected Shortfall (SES) (Acharya and others, 2009, 2010, and 2012), - Distress Insurance Premium (DIP) (Huang and others, 2009 and 2010), - Granger Causality (Billio and others, 2010), - SRISK (Brownlees and Engle, 2011), - Joint Probability of Distress (Segoviano and Goodhart, 2009). - Network analysis and agent-based models relate more closely to the participation approach (Allen and others 2011; Espinosa-Vega and Solé, 2011; OECD, 2012). Haldane and Nelson (2012) highlight non-linearity and unpredictability from networks. - Only a few models estimate multivariate (firm-by-firm) dependence through closed-form specification or simulation of joint probabilities; none include a structural definition of default risk prior to the proposed framework. ### II. METHODOLOGY ### Motivation for Systemic Contingent Claims Analysis (Systemic CCA) - Gap: lack of forward-looking, market-implied multivariate measures of joint default risk with structural default modeling. - Proposal: Systemic CCA — a forward-looking framework for quantifying systemic risk from market-implied interlinkages between financial institutions. - Core idea: view a sample of firms as a portfolio of individual expected losses (with firm-specific risk parameters) and model their joint tail risk via a multivariate density of combined expected losses. - Systemic CCA identifies endogenous linkages affecting joint expected losses during stress, accounting for time-varying dependence. ### Two-step estimation architecture 1. Estimate each firm’s expected losses (and associated change in existing capital levels) using an enhanced form of Contingent Claims Analysis (CCA), which values expected losses as implicit put options. 2. Assume these firm-level expected losses follow a Generalized Extreme Value (GEV) distribution and combine them using a non-parametric dependence measure to derive joint expected losses as a multivariate conditional tail expectation (CTE). ### Main features of Systemic CCA (Table 3) - Model description: - Extends risk-adjusted balance sheet model (CCA) to determine systemic risk magnitude from interlinkages between institutions based on time-varying likelihood of joint declines of implied asset values below the debt-driven “default barrier.” - Quantifies individual institutions’ contributions to systemic solvency risk and assesses spillover risks as contingent liabilities. - Treatment of solvency risk: - Stochastic capital assessment based on expected losses embedded in equity prices and implied asset value, asset volatility, and debt service obligations within risk horizon; relies on option-pricing models to assess maturity mismatches and leverage impacts on capital adequacy. - Treatment of channels of systemic risk: - Estimates non-linear, non-parametric dependence structure between sample firms so linkages are endogenous and dynamic. - Application/Data requirements: - Appropriate when prudential data access is limited but market information is available. - Minimal supervisory data: accounting information on outstanding liabilities, market data on equity and equity option prices, market rates, and macro data for satellite stress-test models. - Strengths: - Integrates market-implied expected losses and endogenizes loss-given-default (LGD) in a multivariate joint-default specification. - Flexible: quantifies time-varying individual contributions to systemic solvency risk and can price a systemic risk charge. - Weaknesses: - Requires assumptions on option-pricing model specification; technique is complex and resource-intensive. ### CCA foundations and enhancements - CCA principles applied: - Liabilities derived from assets; assets follow a stochastic process; liabilities have different priorities. - Equity modeled as an implicit call option; risky debt as default-free debt less an implicit put option capturing expected losses. - Option-pricing enhancements: - Use Gram-Charlier expansion of asset-change density or jump diffusion processes to address BSM shortcomings and capture non-linearities and stochastic volatility. - Other extensions: stationary leverage ratios (Collin-Dufresne and Goldstein, 2001) and stochastic interest rates (Longstaff and Schwartz, 1995). - Expected loss valuation: - Expected loss of a financial institution equals the implicit put option value determined by debt duration, firm leverage, and asset-value volatility. ### Dependence modeling and aggregation - Challenge: simple summation of individual implicit put option values assumes perfect correlation; real aggregation requires the dependence structure of individual expected losses. - Limitations of correlation: - Traditional pair-wise correlation is ill-suited for joint extremes and “fat tails,” since correlation only fully characterizes dependence under elliptical joint distributions (rare in practice). - In stress, default risk is skewed and exhibits excess skewness, excess kurtosis, and non-linear amplification of losses. - Approach: - Model marginal distributions of firm expected losses as fat-tailed within the domain of the GEV distribution. - Combine marginals with time-varying non-parametric dependence to generate a multivariate distribution of joint expected losses. - Iterative estimation over a pre-specified time window; frequency of updates driven by data periodicity. - Outputs: - Point estimates of joint expected losses as multivariate CTE. - Quantification of marginal contribution of specific institutions to joint tail risk at various statistical confidence levels (the contribution is derived as the partial derivative of the multivariate density relative to changes in the relative weight of each bank’s univariate marginal at the specified percentile). ### Positioning relative to other measures - Compared to MES/SES and SRISK: - MES/SES generate empirical linear/bivariate dependence measures; SRISK provides a closed-form specification using DCC-GARCH for correlations but remains different from multivariate density estimation. - Systemic CCA provides: - A multivariate density framework allowing determination of marginal contribution to concurrent changes in both severity of systemic risk and dependence structure across any combination of institutions for any level of statistical confidence and point in time. ### Systemic CCA capabilities for stress testing and capital assessment - Can generate closed-form solutions for market-implied estimates of capital adequacy under stress scenarios. - Links macroeconomic paths and bank-specific income/loss components (net interest income, fee income, trading income, operating expenses, credit losses) to changes in market-implied expected losses (implicit put option values). - Derives market-implied joint capital need from expected losses relative to current (core) capital levels and escalation during extreme market stress at very high statistical confidence levels (expressed as “tail risk”). (footnote 15) ### Model objectives reiterated - Two essential goals: - (i) Measure the extent to which an institution contributes to systemic risk (aligned with the contribution approach). - (ii) Use such a measure to price the potential public sector cost of assisting an institution if it faces capital need. *Italic: Excerpted from the content unit _wp1354 - References* ### 1. Step 1 – Calculating expected losses from contingent claims analysis (CCA) using option ### 1. Step 1 – Calculating expected losses from contingent claims analysis (CCA) using option pricing ### CCA / BSM framework — core concept - CCA generalizes option pricing theory (OPT) pioneered by Black and Scholes (1973) and Merton (1973 and 1974) to the corporate capital structure context; when applied to credit risk it is commonly called the Black-Scholes-Merton (BSM) or Merton model. - Equity holders of leveraged firms are modeled as holding a call option on firm value after outstanding liabilities are paid; conversely, expected losses to creditors are valued as an implicit put option on the firm’s asset value with strike equal to the present value of promised payments on risky debt. - Key balance-sheet identities: - Market value of assets A(t) = market value of equity E(t) + risky debt D (maturing at T). - Default occurs if asset value falls below the bankruptcy level (default barrier) defined as the present value of promised debt payments B discounted at the risk-free rate. - Conventional expected-loss accounting: expected loss = PD × LGD × EAD (probability of default × loss given default × exposure at default). ### Risk-adjusted (CCA) bank balance sheet vs traditional accounting - Traditional accounting balance sheet: - Assets: cash, reserves, loans, credits, other exposures (accounting assets). - Liabilities: debt and deposits, book equity; changes in accounting assets change book equity one-for-one. - Risk-adjusted (CCA) balance sheet: - Assets: implied market value of assets A (cash, reserves, implied market value of “risky” assets). - Liabilities: “risky” debt D (= default-free value of debt and deposits minus expected losses to bank creditors) and observable market equity E (= market capitalization). - Under CCA: - Declines in asset value increase expected losses to creditors and lead to less-than-one-to-one declines in market equity; the equity change depends on severity of distress, leverage, and asset volatility. - Expected loss to creditors becomes a “risk exposure” on the risk-adjusted balance sheet and is linked to market-implied valuations (including government guarantees and investor risk appetite effects). ### Valuation of expected losses as an implicit put option - The present value of market-implied expected losses is valued as an implicit European put option with strike equal to the present value of debt B and underlying asset A(t). - The put option value increases with the probability that implied asset value will fall below the default barrier over horizon T–t; this probability is driven by the level and volatility of implied asset value reflected in equity and equity option prices conditional on capital structure. - Model relations (as presented): - Expected-loss put option expression (Equation (1) in source) relates put value to A(t), B, r, σ, and cumulative normal Φ. - Relation linking equity volatility Eσ, equity price Et, asset value A(t), and asset return volatility (Equation (2) in source) is used to connect observable equity variables to implied asset volatility. - Leverage enters via a sensitivity parameter (Equation (3) in source) involving ln(A/B), σA, and time-to-maturity terms. - Observables and notation used: - Eσ = observable equity volatility. - Et = equity price. - r = risk-free discount rate. - Φ = cumulative standard normal distribution. - Duration T–t and leverage drive sensitivity of put value to asset-value changes. - Practical estimation note: - Asset value and asset volatility can be derived from equity and equity option prices; absence of listed equity data can be addressed with peer-group analysis and historical balance-sheet scaling (example procedure described for non-listed firms). ### Extensions and refinements (Box 1) - Limitations of the plain Merton model (constant volatility, lognormal diffusion, no skew/kurtosis) motivate closed-form extensions that preserve analytical tractability. - Gram-Charlier (GC) expansion: - Introduces correction terms for skewness (γ1) and kurtosis (γ2) into the put price while keeping the diffusion framework. - GC requires estimating two additional parameters (skewness and kurtosis). - Jump-diffusion specification: - Introduces a Poisson jump process with intensity λ (average number of expected jumps per unit time). - Jump size follows a log-normal distribution with average jump size φ and volatility ν; calibration can use an estimation window with τ-number of observations (example τ = 120 days). - Put value is written as an infinite series summing contributions conditional on k jumps. - Asset volatility consistent with jump diffusion becomes σ_A^2 = σ_A^2 + kν^2 (as in the source formulation) and the risk-free rate is adjusted with terms involving λ and φ. - Other modeling options mentioned: - Stochastic volatility models (Heston 1993; Heston and Nandi 2000), GARCH, copulas, Levy processes, neural nets, non-parametric methods, and lattice/binomial trees (Cox and others, 1979). - Empirical findings cited: introducing stochastic volatility yields most pricing improvement; introducing jumps leads to smaller improvements. ### Estimation practices and caveats - Alternative estimation avoids the traditional two-equations-two-unknowns approach by deriving implied asset value directly from the state-price density (SPD) estimated from option prices using the Breeden and Litzenberger (1978) method plus a semi-parametric Black-Scholes specification (Aït-Sahalia and Lo, 1998). - This uses the second derivative of call price with respect to strike across options with identical maturity, interpolating and extrapolating the implied volatility surface via nonparametric local polynomial regression when strikes are discretely spaced. - Sources of estimation uncertainty and biases: - Distributional assumptions, derivation of implied assets and asset volatility, default barrier assumptions, and potential failure to capture relevant economics can bias expected-loss estimates. - Equity prices may reflect non-fundamental factors (shareholder dilution, trading behavior, “flight to quality”) that complicate interpretation—e.g., rapid declines in market capitalization during crises may not solely signal future solvency risk. - Reconciliation: the risk-adjusted balance sheet converges to the accounting balance sheet if asset volatility is zero (expected loss to creditors declines to zero and equity equals book equity). ### Practical use and implications for policy and stress testing - CCA enables integrated modeling of default risk to: - Quantify impact on bank borrowing costs of higher/lower equity levels. - Measure the impact of changes in investor risk aversion on bank valuations and expected losses. - Assess implications of public sector support (implicit/explicit guarantees) on market-implied capital assessments and funding costs. - Empirical application in stress-testing frameworks: - More advanced option-pricing methods (GC, jump diffusion) have been applied in Systemic CCA within IMF FSAP stress tests for multiple countries. - Where market data are unavailable for non-listed institutions, peer-group scaling and historical volatility estimation can be used to derive implied assets and asset volatilities. *From: 1. Step 1 – Calculating expected losses from contingent claims analysis (CCA) using option pricing* ### 2. Step 2 – Estimating the joint expected losses from default risk ### 2. Step 2 – Estimating the joint expected losses from default risk ### Overview - Individually estimated expected losses (series of put option values per firm) are combined to determine system-wide default risk. - Expected-loss distributions are assumed “fat-tailed” in line with extreme value theory (EVT); this informs estimation of the dependence function and a multivariate extreme value framework (Systemic CCA). - Tail risk measures such as conditional Value-at-Risk (VaR) or expected shortfall (ES) are used to gauge systemic solvency risk at a chosen statistical confidence level. ### (i) Estimating the marginal distributions of individual expected losses - Individual expected losses are modeled using EVT and the Fisher-Tippett-Gnedenko theorem; observed series: - X = (EmP1(t), ..., EmPn(t)) denotes i.i.d. random observations of expected losses (a total of n daily put option values up to time t), each estimated over a rolling window of τ observations with periodic updating (example: a daily sliding window of 120 days). - Limiting extremal types and GEV specification: - Extremal types: Gumbel (EV0), Fréchet (EV1), negative Weibull (EV2). - Unified GEV probability density and cumulative distribution functions are given (equations (10) and (11)). - Univariate marginal density converging to GEV: - jth density: f̂_j(x) as specified in equation (12) with conditions: - 10_j j j x ξ μ σ + − >, scale parameter 0_j σ >, location j μ, shape 0_j ξ ≠. - Moments (preserved exactly as presented): - mean: 1 h σ σ μ ξ ξ − + given conditions in equation (13). - variance: (22 21 2) h h σ ξ − given conditions in equation (14). - skewness: (3 312 1 32 2 21 32 hhh h hh − + −) given conditions in equation (15). - kurtosis: (24 413211 2 2 21 463 3 hhhhhh hh − + − − −) given conditions in equation (16). - with hp = Γ(1 − p ξ) for p ∈ {1,...,4}, Euler’s constant γ, Riemann’s zeta function ζ(t), and gamma function Γ(t). - Estimation method: - Moments and parameters estimated via numerical iteration using maximum likelihood (ML) over rolling windows (e.g., τ observations), maximizing L(μ,σ,ξ) = ∏_{i=1}^n h(x_i; μ,σ,ξ). - The linear combinations of ratios of spacings (LRS) estimator serves as an initial value. - Note: ML estimator fails for 1 ξ ≤ − because the likelihood has no global maximum in this case (a local maximum near the initial value can be attained). ### (ii) Estimating the dependence structure of individual expected losses - A non-parametric multivariate dependence function is defined by extending the bivariate logistic method of Pickands (1981) to multivariate and adjusting margins per Hall and Tajvidi (2000). - Dependence function specification (equation (17)): - Uses average marginal density estimates n̂^{-1} ∑ n_{·j} y_{ij} and factor weights ω_j with constraints: - ω_j ∈ [0,1] and mj ω_{·} relationships given (see equation (18) for the (m−1)-dimensional unit simplex). - γ(·) is a convex function on [0,1] with γ(0)=γ(1)=1 corresponding to complete dependence and mutual independence at extremes. - Estimation: - γ is estimated iteratively over rolling windows (e.g., τ = 120 days), subject to optimization on the unit simplex S_m (equation (18)). - Contrast with copula approaches: - This specification departs from a single time-invariant copula dependence parameter by explicitly estimating a multivariate dependence function that captures varying degrees of coincidence across series. ### (iii) Estimating the joint distribution of expected losses - Margins and the dependence function are combined to form a multivariate extreme value distribution (MGEV) over the same estimation period. - Joint cumulative distribution function (equation (19)): - G(·) = exp(−∑_{j=1}^m [·]^{−1/ξ_j} • γ(·)) (as per the specified functional form). - Joint density function (equation (20)) is given with parameters μ_j, σ_j, ξ_j, ω_j and is estimated by maximizing the joint likelihood over all parameters simultaneously: - Maximize L(θ) = ∏_{t=1}^m g_t(·; θ) with θ = (μ,σ,ξ). - Log-likelihood ln L = ∑_{j=1}^m ln g_j, and MLE θ̂ = arg max_{θ∈Θ} ln L (equation (21)). ### (iv) Estimating a tail risk measure of joint expected losses (ES / conditional VaR) - ES definition and properties: - ES_a(X) = E[X | X > VaR_a(X)] for continuous X; represented in integral form in equation (22). - ES is a coherent risk measure satisfying axioms: - H1. Subadditivity: ϑ(X+Y) ≤ ϑ(X) + ϑ(Y). - H2. Monotonicity: X ≤ Y ⇒ ϑ(X) ≤ ϑ(Y). - H3. Positive Homogeneity: ϑ(λX) = λϑ(X) for λ > 0. - H4. Translation Invariance: ϑ(X + c) = ϑ(X) + c. - ES is presented as an improvement over VaR, which can be “incoherent.” - Multivariate ES (equation (23)): - ES_{a,τ}(·) computed via integrals over the joint distribution G and its quantile G^{-1}_a, using the dependence structure γ and MGEV. - The joint VaR condition stated in equation (24) for the 0.95 example: sup { z : Pr(G^{-1}_a ≤ z) ≥ 0.95 }. - Point estimate of joint potential losses for m firms at time t (equation (25)): - Given explicitly as a function of estimated parameters ξ̂, μ̂, σ̂, ω and τ. ### (v) Estimating the individual contribution to joint expected losses - Individual contribution determined via cross-partial derivative of the joint distribution of expected losses (second-order cross-partial of inverse joint density G^{-1} with respect to dependence γ and marginal severity y_{t,j}). - Joint ES can be written as a linear combination of individual ES values ES_{t,j,a,τ} with weights ψ_{a,τ}^{(j)}: - ES_{a,τ} = ∑_{j=1}^m ψ_{a,τ}^{(j)} ES_{t,j,a,τ} (equation (26)). - Relative weight / marginal contribution ψ_{t,τ}^{(j)} defined (equation (27)) as: - ψ_{t,τ}^{(j)} = ∑_{s=1}^m ∂^2 G^{-1}_a / (∂γ ∂y_{t,j}) evaluated at the specified percentile, subject to ∑_j ψ_{t,τ}^{(j)} = 1. - Use cases: - Identifies each institution’s marginal impact on systemic tail losses. - Can be used to evaluate the effect of supervisory remedial actions (e.g., reducing a bank’s contribution to systemic liquidity risk). ### III.A Extensions — Price-based macroprudential measure for systemic risk - Motivation: - Joint expected losses from Systemic CCA can inform macroprudential instruments targeting SIFIs / LCFIs / “too-big-to-fail” institutions. - Insurance premium concept: - An insurance policy indemnifying a firm’s contribution to systemic risk can be priced actuarially using the firm’s marginal contribution to joint ES relative to aggregate default barrier (sum of individual default barriers). - Insurance premium formulation (equations preserved exactly): - Hazard-rate based relation (equation (28)): - φ_{t,a,τ}^{(j)} = − ln(1 − B_{t}^{−1} ∑_{·} ψ_{·} ES_{·}) ≈ − ln(1 − ... ) with the expression in the source (equation (28) exact form preserved). - Defined as ratio between average marginal contribution of each firm to average expected shortfall (over multiple periods, e.g., last four quarters if expected losses estimated daily) and average discounted present value of total liabilities of all sample firms: ∑_j e^{−rT_t} B_{t,j}. - Survival probability and hazard relation (equation (29)): - φ(t) = 1 − exp(−∫_0^t u(s) ds) with the exponential survival formulation preserved. - Converting premium to monetary insurance cost on short-term liabilities (equation (30)): - Insurance_{ST,t,j} = φ_{t}^{(j)} × B_{ST,avg} converted with the ln and 110,000 scaling factor preserved as shown in equation (30). - Policy considerations and caveats: - Premium would internalize own-default externalities and aim to change firm behavior rather than primarily fund ex post public costs. - Implementation risks: - Additional burden on financial sector when capital is scarce; careful calibration and long implementation periods needed so credit supply is not impeded. - Institutionalizing systemic importance could exacerbate moral hazard and a sense of entitlement to government support. - Questions remain about ex ante vs ex post charging and destination of proceeds (special funds vs general revenue). *Source: _wp1354 - 2. Step 2 – Estimating the joint expected losses from default risk* ### Box 2). The financial crisis that started in 2007 is a stark reminder of how public sector ### _wp1354 - Box 2). The financial crisis that started in 2007 is a stark reminder of how public sector ### Sovereign–bank negative feedback and contingent liability channel - Public sector support measures to large financial institutions can result in considerable risk transfer to the government, amplifying the emphasis on long-term fiscal sustainability. - A decline in the market value of sovereign debt: - weakens implicit sovereign guarantees to systemically-important financial institutions (large banks in particular), - raises default risk in the financial sector overall (increases the cost of borrowing), - increases likelihood of downgrades to financial institutions following sovereign downgrades, establishing an upper ceiling to unsecured funding ratings. - The linkage is accentuated by the lender–borrower channel when financial institutions hold large exposures to public sector entities. - Diagrammatic feedback elements (as described): - Lower mark-to-market (MTM) in value of all government bonds held by local banks → Increase in bank funding costs → Erosion in potential for official support/bailout → Increase in contingent liabilities. - Lower MTM in value of all government bonds held by foreign banks → Increase in contingent liabilities → Rise in counterparty credit risk → Similar sovereigns come under pressure. ### Market-implied contingent liabilities via Contingent Claims Analysis (CCA) - CCA combines market-implied expected losses from equity market and balance sheet information with CDS contract information to estimate government contingent liabilities. (See footnote 50 and 51 context.) - Heuristic derivation of market-implied government guarantee for large banks: - Government guarantee ≈ difference between total expected loss (the value of a put option E Pt) and the value of an implicit put option derived from the bank’s CDS spread (CDS Pt), assuming equity values remain unaffected near default. - The put implied by CDS, CDS Pt, reflects expected losses associated with default net of any financial guarantees (residual default risk on unsecured senior debt). - The ratio alpha defined in equation (35): - alpha(t) = 1 − CDS_Pt / E_Pt - alpha(t) defines the share of expected loss covered by implicit (or explicit) government guarantees that depress CDS spreads below the level warranted by equity-implied default risk. - alpha represents the fraction of default risk covered by the government; (1 − alpha) E_t P_t is the risk retained by an institution and reflected in the senior CDS spread. - Caveats: - Estimation depends on assumptions that influence assessment of likelihood of public sector support, especially in extreme stress. - Differences between equity-implied and CDS-implied put option values can reflect modeling choices, breakdowns of efficient asset pricing in illiquidity, capital-structure impacts of government interventions (e.g., equity dilution), recovery assumptions, and differing risk horizons. - An adjustment for “basis risk” is used (referenced to equation (30) in source). ### Key mathematical and technical elements preserved from the source - Specification (paraphrased with preserved numeric elements and notation references): - CDS spread in basis points: CDS_st, and its rearranged specification under risk-neutral measure leading to expressions involving terms like exp(−rT t), Be, Dt, B, and constants such as 10 000 and 11 in the algebraic forms (see equations (31)–(34) in source). - Default probability at time t under constant hazard rate approximation: integral expression leading to notation φ_CDS_st and implied yield to maturity yr = φ − r. - Relation: rT t CDS_st Be P DLGD −− − − = × over period 1Tt− = 1 (textual preservation of source expression). - Recovery-at-face value (RFV) vs. recovery-at-market-value (RMV) effects: if Dt > B, RFV drops below RMV → CDS spread CDS_st becomes drops below the bond spread; below-par bond price pushes up implied recovery rate of CDS, causing CDS spread mark-up over comparable bond spreads. - Footnote context: basis approximation using Moody’s KMV FVOAS and FVCDS; adjustment factor close to unity for most cases, with a few cases in a 20 percentage point range (0.9 to 1.1). - Use of European put option assumption noted as possibly overstating expected losses relative to put option derived from CDS spreads (no early exercise). ### Systemic CCA: linking sovereign and banking sector balance sheets - Systemic CCA replaces individual expected losses with contingent-liability measures (put values) to derive systemic risk from joint contingent liabilities. - Sovereign CDS spread expressed as: - sov_CDS_st = 1 ln 110,000 sov rT t sov Pt st Tt Be −− = − × (preserved textual fragments from source). - Sovereign implicit put option and sovereign asset calibration: - sovereign put value expressed in terms of implied sovereign assets A_sov_t, asset volatility σ_sov_A, default barrier B_sov, risk-free rate r, and duration T−t with Φ terms (normal CDF) as per source equations. - The term structure of sovereign spreads at tenors 1, 3, 5, 7, and 10 years is used to determine combination of sovereign assets and asset volatility that match observed credit spreads at chosen maturity term. - Sovereign asset value decomposition (preserved form and components): - sov_At = R + PV(primary fiscal surplus) − bank_t P_bank α + Other (where R = foreign currency reserves; net fiscal assets = present value of primary fiscal surplus (PS); implicit and explicit contingent liabilities to banking sector = bank_t P α; Other = remainder items). - Bank contingent liabilities: - Bank put option value (bank put) expressed as function of implied bank assets, asset volatility, and time horizon (explicit notation preserved in source). - Full specification of sovereign CDS spread includes R, PS, bank contingent liabilities (α times bank put), and Other. ### Feedback mechanisms and destabilization channels highlighted - If sovereign spreads increase: - Value of α decreases (potential for sovereign guarantees decreases) → bank spreads increase. - Implicit bank put option value increases as value of bank holdings of government debt decreases. - Bank default barrier may increase due to higher borrowing costs as the premium δ increases. - Bank CDS spread equation (preserved structure) includes retained fraction (1 − α) of bank default risk plus constant δ reflecting add-on to CDS premium; if modeled bank CDS falls below observed bank CDS, δ captures extra premium from high sovereign spreads increasing expected bank losses. ### Integrated market-implied capital assessment (CCA and Systemic CCA) - CCA provides risk-adjusted (economic) balance sheet approach and its multivariate Systemic CCA extension offers an integrated market-based assessment of individual and system-wide solvency. - Two broad indicators: - Market-implied capital adequacy ratio (MCAR) = Market Capitalization / Implied Assets. - Expected Loss Ratio (EL Ratio) = Expected Losses (from CCA) / Market Capitalization. - System-wide (aggregate) counterparts: - Aggregate MCAR = Aggregate Market Capitalization / Aggregate Implied Assets. - Joint EL Ratio = Joint Expected Losses (from Systemic CCA) / Aggregate Market Capitalization (example percentile: 95th). - MCAR as market analogue to regulatory capital adequacy (distinct from risk-weighted assets). CCA-based measures can be used to identify market-implied capital shortfalls relative to regulatory definitions. *Source: Excerpt from the referenced IMF working paper content unit provided.* ### Section V). ### _wp1354 - Section V) ### Sensitivity of fair value credit spreads and market-implied default risk - Fair value credit spread (in basis points) is specified as a constant hazard rate and formalized in equation (37). Individual and joint expected-loss-based periodic default probabilities are given in equations (39) and (40), with implications for implied periodic default probabilities and for market-implied capital adequacy measures (MCAR). - Non-linear dynamics highlighted: - Higher expected losses → lower implied asset value and higher implied asset volatility → higher equity risk premium → EL ratio rises and MCAR decreases → higher fair value credit spread. - The framework links individual and joint expected losses to fair value credit spreads and to individual and joint periodic default probabilities (see equations (37)–(40)). ### Market-implied capital shortfall and reconciliation with prudential solvency standards - Market-implied capital shortfall (in the CCA context) is defined as the amount of capital required to maintain solvency conditions under stress irrespective of changes in expected losses, net of any existing common equity buffer. - This definition allows: - A risk-based assessment of capital adequacy to be reconciled with prudential solvency standards. - Conversion of market-based measures (EL, MCAR) into capital shortfall estimates comparable to balance sheet-based analysis. ### Systemic CCA for stress testing: framework and scenario construction - Systemic CCA projects systemic solvency risk by forecasting firm-specific market-implied expected losses over a selected forecast horizon and combining them to define system-wide solvency risk. - Three firm-specific CCA-based measures used empirically: (i) contingent liabilities as the share of expected losses potentially transferred to the public sector if a systemically relevant firm were to fail (see Section III.B); (ii) capital shortfall based on expected losses in excess of existing common (core) equity Tier 1 capital above the regulatory minimum (see Section III.C); (iii) capital shortfall based on the market-implied capital adequacy ratio (MCAR) generated from the change in market capitalization relative to asset value under the impact of expected losses (see Section III.C). - Linking expected losses to macro-financial scenarios: - Empirical (satellite) and theoretical (structural) models can specify macro-financial linkages of expected losses under stress scenarios. - Satellite model: historical sensitivity of market-implied expected losses (or option-pricing elements) estimated on macroeconomic and bank-specific variables using econometric approaches (e.g., dynamic panel regression). - Structural model: implied asset values adjusted by forecasts of net operating income, with re-estimated implied asset volatility, to derive revised put option values and changes in market-implied expected losses relative to maturing debt obligations. ### Multivariate aggregation and dependence modeling - Joint contingent liabilities and potential capital shortfall are derived by estimating the multivariate density from each bank’s marginal distribution of forecasted expected losses and their dependence structure. - If forecasted expected-loss series are short (e.g., quarterly values over a five-year horizon), historical expected losses may be appended to reach a sufficient number of observations for reliable estimation of univariate marginals and dependence. - Emphasis on capturing: - Variability of risk factors and their dependence under all plausible scenarios. - Non-linear dependence and joint tail risk (VaR and Expected Shortfall (ES) approaches). - Time-varying correlations that can be weak in normal times and very high under distress, producing excess skewness and excess kurtosis with implications for tail risk measurement. ### Empirical application: U.S. financial system (IMF FSAP context) - Sample and data: - The model was estimated on market and balance-sheet information of 33 large commercial banks, investment banks, insurance companies, and special purpose financial institutions using daily data between January 1, 2007 and end-January 2010. - Sample period (specific charts): 01/03/2007-01/29/2010. - Sample firms (illustrative listing from the sample): Bank of America, J.P. Morgan Chase, Citigroup, Wells Fargo, Goldman Sachs, Morgan Stanley, Bear Stearns, Lehman Brothers, PNC Financial Services Group, U.S. Bancorp, SunTrust Banks, BB&T Corp., Regions Financial Corporation, Fifth Third Bancorp, KeyCorp, Bank of New York Mellon Corp., State Street Corp., Northern Trust Corp., CIT, Ameriprise, Fannie Mae, Freddie Mac, Sallie Mae, AIG, Metlife, Prudential, Hartford, Allstate, Principal, Travelers, Genworth, Aflac, Lincoln. - Key inputs and parameters used in empirical estimation: - daily implied asset values derived as the expectation over the state price density (SPD) using equity option data (from Bloomberg), - default barrier estimated for each firm based on quarterly financial accounts (from Moody’s KMV CreditEdge), - the risk-free rate of interest r=3.0 percent, - a one-year time horizon T=1, - one-year credit default swap (CDS) spreads on senior debt (from MarkIt). - Default barrier = total short-term debt plus one-half of long-term debt (consistent with Moody’s KMV CreditEdge). - SPD calculated using daily European-style equity call option price data with a time-to-maturity of three months; outputs were expected losses (implicit put option value over a one-year horizon) and associated contingent liabilities (alpha-value times implicit put option). - Multivariate distribution estimated over a rolling window of 60 working days (i.e., three months) with daily updating. - Empirical findings and diagnostics: - Alpha-value dynamics (Figure 5): median and inter-quartile range of alpha-values across 29 sample firms with significant contingent liabilities show a secular increase and decreasing dispersion, indicating growing implicit financial sector support; sudden declines in alpha-values around April 2008, October 2008, and May 2009 reflect reduced market confidence in explicit or implicit public sector support amid greater alignment of falling market capitalization and rising CDS-implied default risk. - Aggregate Systemic CCA estimates (Figure 6): market-implied total contingent liabilities expressed in percent of real GDP, with the multivariate density generated from univariate marginals conforming to the Generalized Extreme Value Distribution (GEV) and a non-parametrically identified time-varying dependence structure; marginal severity and dependence estimated over a window of 60 working days with daily updating. - Empirical progression: from summation of individual contingent liabilities derived from implicit put option values to the multivariate distribution and calculation of institution-specific contributions to joint contingent liabilities and systemic risk surcharges (as specified in equation (30)). ### Methodological considerations and benefits/drawbacks - Benefits of valuation/distributional approaches: - Allow characterization of scenarios and combinations of risk factors without re-estimating satellite models. - Enable sensitivity analysis and probabilistic capital assessments via VaR or Expected Shortfall (ES). - Drawbacks and caveats: - Validity of valuation models may be undermined by distress events and rare, non-recurring realizations beyond historical precedent. - Statistical apparatus underlying conventional asset pricing may fail to capture extreme, sudden events (e.g., recent financial crisis episodes). - Reliable estimation requires a sufficient number of observations for univariate marginals and dependence. *Source: _wp1354 - Section V).* ### Appendix 3). There are two 50 ### _wp1354 - Appendix 3). There are two 50 ### Divergence of Total Expected Losses and Contingent Liabilities (Figure 6) - Total expected losses (area) and contingent liabilities (lines) began to increasingly diverge during the beginning of the financial crisis. - Each reached their highest values between the periods just after the collapse of Lehman Brothers in September 2008 and end-July 2009. - The persistent difference between expected losses and contingent liabilities suggests markets expected that on average more than 50 percent of total expected losses could have been transferred to the government in the event of default. - The sum of individual contingent liabilities: - peaks at more than 40 percent of GDP at the end of February 2009, - averages 2.5 percent of GDP over the entire sample period. - Controlling for the dependence structure using the aggregation technique underpinning the Systemic CCA framework (see equations (19) and (20) above) caused the aggregate measure of contingent liabilities (the median value (50th percentile) of the multivariate distribution of contingent liabilities) to drop significantly (orange and red dotted lines in Figure 6). ### Estimating the Systemic Risk from Joint Contingent Liabilities (Figure 7) - Systemic risk was derived from the multivariate distribution of contingent liabilities estimated on a daily basis over a 60-day rolling window. - Systemic risk from contingent liabilities: - was considerable during the financial crisis, - increased sharply after the collapse of Lehman Brothers. - Time pattern showed spikes as early as April 2008 (post-Bear Stearns bailout). - Extreme tail risk (the 95th percentile ES) of expected losses transferred to the government: - exceeded nine percent of GDP already in April 2008, - almost reached 20 percent of GDP in October 2008. - Interpretation: market prices implied joint contingent liabilities of 20 percent of end-2009 GDP with a probability of less than five percent over a one-year horizon during exceptional systemic distress. - The magnitude of such tail risk: - dropped to under two percent of end-2009 GDP during the remainder of 2008, - rose again in May 2009. - Notes on Figure 7: - Sample period: 01/03/2007-01/29/2010, daily observations of individual put option values conditional on the endogenous alpha-value of implicit guarantees of 36 banks, insurance companies, and other financial institutions. - The red line shows the expected shortfall (ES) for the entire sample at a 95th percentile threshold within a confidence band of one and two standard deviations (grey areas). - The multivariate density is generated from univariate marginals that conform to the Generalized Extreme Value Distribution (GEV) and a non-parametrically identified time-varying dependence structure. - Marginal severity and dependence are estimated over a window of 60 working days (with daily updating). ### Individual Contributions to Systemic Risk from Contingent Liabilities (Table 6, Figure 8) - Failed and bailed-out firms were on average the largest contributors to systemic risk from joint contingent liabilities, especially prior to the collapse of Lehman Brothers. - Percentage share by group (selected entries from Table 6; in percent of average expected shortfall at the 95th percentile during each time period): - Banks: Pre-Crisis July 1, 2007—Sept. 14, 2008 = 15.6; Total Period April 1, 2007—Jan. 29, 2010 = 27.2 - Insurance companies: Pre-Crisis = 13.9; Total Period = 15.0 - Other, non-bank financial institutions: Pre-Crisis = 3.0; Total Period = 14.7 - Failed (or bailed-out) financial institutions: Pre-Crisis = 69.5; Total Period = 43.1 - Dynamics: - Pre-crisis (up to September 14, 2008), nine institutions that later failed or were rescued contributed 69.5 percent. - After September 15, 2008, failed institutions dropped out and their contribution declined while large banks increased their share to about one-third of total systemic risk. - Contribution of insurance companies remained low during the height of the credit crisis before rising to nearly 24 percent on average during the second half of 2009. - Comparison to naïve measures: - Ignoring dependence structure and summing contingent liabilities would have: - overstated the tail risk contribution of failed/rescued institutions by 43.5 percentage points on average over April 2007—end-January 2010, - understated contribution of other banks by 19.5 percentage points and insurance companies by 10.2 percentage points. - Using share of reported total liabilities is more consistent with Systemic CCA groupwise contributions, especially for failed/rescued institutions and insurance companies. - Notes on Figure 8: - Sample period: 01/03/2007-01/29/2010, daily observations conditional on endogenous alpha-value of 36 institutions. - Chart shows group-wise contribution to ES at 95th percentile; multivariate density uses GEV marginals and non-parametric time-varying dependence; marginals and dependence estimated over 60 working days with daily updating. ### Calculating a Systemic Risk Surcharge for Contingent Liabilities (Table 7) - A systemic risk surcharge was modeled as an insurance contract covering expected cost of public sector interventions. - Premium rate calculation: - Based on systemically-based dollar losses of each institution as if a five percent tail event had occurred relative to aggregate outstanding liabilities of all sample firms (see equation (30)). - Average marginal contribution of each firm to joint contingent liabilities divided by average of discounted present value of total liabilities of all firms over the same period to derive the hazard rate. - Findings: - Annual systemic surcharge for systemically important financial institutions of about 50 basis points on average. - Pre-crisis (July 1, 2007—Sept. 14, 2008) reasonable average surcharge: 39 basis points per year. - At a higher statistical confidence level (95th percentile), surcharge would have reached: - 60 basis points shortly before the collapse of Lehman Brothers, - 317 basis points shortly after the collapse. - Controlling for the share of insured deposits reduces the total default barrier B by 15 percent on average, decreasing the estimated systemic risk surcharge accordingly. - Table 7 entries (selected): - Pre-Crisis July 1, 2007—Sept. 14, 2008: Amount (In billion U.S. dollars) = 59; Annual fee/surcharge (In basis points) = 39 (50th percentile) - Crisis Sept. 15—Dec. 31, 2008: Amount = 432; Annual fee/surcharge = 479 (95th percentile) - Total Period April 1, 2007—Jan. 29, 2010: Amount = 744; Annual fee/surcharge = 214 (50th percentile); 142 (95th percentile) - Note: Given time-variation, surcharge could be augmented with counter-cyclical properties by combining estimates over different estimation periods and percentile levels. ### Empirical Application: Stress Testing Systemic Risk from Expected Losses in the U.K. Banking Sector - Application context: - Systemic CCA used as a market-based top-down solvency stress testing model in the IMF’s FSAP stress test of the U.K. banking sector. - Sample: five largest commercial banks, the largest building society, and the largest foreign retail bank; daily data January 3, 2005—end-March 2011. - Non-linearity of individual and joint expected losses (Figures 9 and 10): - Single-firm example: rising default risk increased sensitivity of Market-implied Capital Adequacy Ratio (MCAR) to expected losses; gap between MCAR and Core Tier 1 ratio widened toward the end of sample. - System-wide example: MCAR showed even higher sensitivity to changes in system-wide expected losses; improvement in solvency after March 2009 appeared more protracted across sample firms. - Estimating expected losses under stress using macro-financial linkages: - Two methods: - IMF satellite model: dynamic panel regression with macro variables (short-term interest rate [+], long-term interest rate [-], real GDP [-], unemployment [+]) and projected individual bank performance from RAMSI (net interest income [+], operating profit before taxes [-], credit losses [+], leverage [+], funding gap [+]). - Structural model: adjust implied assets as at end-2010 by forecasts of operating profit and credit losses from RAMSI, re-estimate implied asset volatility, derive revised put option value. - Forecast horizon: 2011–2015; individual expected losses transposed into capital shortfall assuming increases in expected losses in excess of existing capital buffers constitute a shortfall of Tier 1 capital. - Joint capital shortfall distribution estimated using Systemic CCA with univariate densities and dependence combined into a multivariate distribution (five-year sliding window, monthly updates). - Main outcomes: - Joint capital shortfall at the 50th percentile would have been contained under all adverse scenarios; existing capital buffers sufficient to absorb median joint solvency risks. - Severe double-dip recession scenario had the biggest impact but would have produced: - joint potential capital losses averaging £0.4 billion (equivalent to 0.03 percent of end-2010 GDP) over 2011–15 under baseline conditions, - up to an average 0.12 percent of end-2010 GDP (£1.8 billion) over the forecast horizon under the severe double-dip scenario. - No capital shortfall relative to Basel III transitional hurdle rates or FSA interim capital regime requirements in these central estimates. - Tail-of-tail analysis (five percent “tail of the tail” beyond 95th percentile): - Under extreme adverse scenarios, market-implied capital shortfall could increase substantially as profitability falls and asset quality deteriorates. - Severe adverse scenario could have resulted in average joint capital losses of up to 3.4 percent of 2010 GDP (£50 billion), still below crisis peaks. - Under that scenario, estimated capital loss could have caused an average capital shortfall of between 1.3– (text truncated in source). *Source: IMF content unit as supplied.* ### 1.6 percent of end-2010 GDP relative to the Basel III Tier 1 hurdle rates (and 1.6–1.8 ### _wp1354 - 1.6 percent of end-2010 GDP relative to the Basel III Tier 1 hurdle rates (and 1.6–1.8 ### Key quantitative findings from United Kingdom Systemic CCA stress tests - Joint capital losses under the prolonged slow growth scenario: up to 2 percent of end-2010 GDP (£29 billion). - Translates to an average capital shortfall of up to 0.4 percent of end-2010 GDP (£6.3 billion). - Baseline to mild/severe double-dip (without debt haircuts, from 2013 onwards): potential losses could be zero. - Under a severe double-dip scenario with sovereign and bank debt haircuts in 2011, estimated potential losses: between 6.4–7.1 percent of end-2010 GDP, or £94–104 billion (depending on the use of the structural or the IMF satellite model). - Resulting estimated capital shortfall relative to Basel III Tier 1 hurdles: between 4.4–5.0 percent of end-2010 GDP, or £63–73 billion. - Relative to the FSA interim capital regime Tier 1 requirements: between 4.9–5.6 percent of end-2010 GDP. - Reported earlier estimates: 1.6 percent of end-2010 GDP relative to the Basel III Tier 1 hurdle rates (and 1.6–1.8 percent of end-2010 GDP relative to the FSA interim capital regime Tier 1 requirements), depending on the choice of satellite model. - Probabilities: a severe double-dip recession scenario was assigned a two percent probability; the joint realization of capital losses beyond the 95th percentile (95th percent ES) implied a less than 0.1 percent probability event, given that [0.02(1-0.95)]=0.001. ### Scenario sensitivity and model differences - Outcomes depend critically on: - Timing and adversity of macroeconomic conditions. - Evolution of sovereign risk affecting banks’ government and bank debt holdings. - Choice of satellite model (IMF satellite model versus structural/RAMSI approach). - Treatment of losses from sovereign and bank debt holdings differs by satellite model: - IMF satellite model: losses from haircuts on holdings of sovereign and bank debt are calculated each year and added to estimated overall potential losses. - Alternative (structural/RAMSI-integrated) satellite model: losses from these debt holdings are subtracted from RAMSI-model projected operating profit each quarter, altering implied firm assets and potential losses via an option pricing approach. ### Market-implied joint potential capital loss estimation (methodology and sample) - Sample period: 01/03/2005-03/29/2010 (monthly observations of historical and forecasted joint capital losses for seven sample firms). - Multivariate density construction: - Univariate marginals conform to the Generalized Extreme Value Distribution (GEV). - Dependence structure: non-parametrically identified time-varying dependence. - Estimation windows: - Historical measure: rolling window of 120 working days with daily updating (sample cut-off at end-2010). - Forecasted values: rolling window of 60 months with monthly updating (uses historical dynamics of capital losses as statistical support). ### Distribution of individual bank contributions to systemic solvency risk - Evidence of concentration: with the exception of the first phase of the European sovereign debt crisis (most of 2010 and beginning of 2011), one bank (the maximum of the distribution) often accounted for more than half of solvency risk in the sample. - Summary statistics (Average per time period, percent of joint capital shortfall, sample period 01/03/2005-03/29/2010): - Minimum (Average): 0.6 - 25th percentile (Average): 2.0 - Median (Average): 5.7 - 75th percentile (Average): 17.2 - Maximum (Average): 55.3 ### Methodological caveats and limitations highlighted - Conceptual and practical challenges in assessing systemic risk: - Results highly dependent on assumptions, differences in business models, and regulatory/supervisory approaches across countries. - Complexity of modeling risks loss of transparency; danger of underestimating practical problems in real-life application. - Market-based determination of default risk may be influenced by: - Validity of valuation models (option pricing theory, extreme value measurement, non-parametric dependence specification). - Financial market behavior that may defy statistical assumptions—rare, non-recurring events during crises may not be captured by models assuming stochastic stability. - Parameter sensitivity and estimation risk warrants use of different valuation techniques to substantiate comprehensive risk assessment (e.g., exploring different option pricing models to detect pricing distortions). ### Extensions and further applications proposed - Expand Systemic CCA scale and scope to include other risks: - Adaptation to measure systemic liquidity risk by transforming the Net Stable Funding Ratio (NSFR) into a stochastic measure of aggregate funding risk. - Integration of financial sector and sovereign risk analysis with macro-financial feedbacks to inform monetary and fiscal policy and calibration of stress scenarios. - Use of an economy-wide Systemic CCA for all sectors (financial sector, non-financial corporates, households, government) to generate new measures such as the present value of risk-adjusted GDP. - Existing related model: Systemic Risk-adjusted Liquidity (SRL) model combines option pricing with market information and balance sheet data to generate a probabilistic measure of multiple entities experiencing a joint liquidity event; links maturity mismatch, overall risk profile, and funding stability across firms. *Italicized source attribution: IMF Working Paper content (sample period and tables as provided).* ### Chapter 3, April (Washington, D.C.: International Monetary Fund), available at ### _wp1354 - Chapter 3, April (Washington, D.C.: International Monetary Fund), available at ### Appendix 1. Standard Definition of Contingent Claims Analysis (CCA) - Core 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. - Default definition: - Default occurs when asset value falls below the default barrier (the present value of promised payments on debt). - Repayment of debt is “risky” if there is a chance of default; guarantees convert risky repayment into expected loss (Merton and Bodie, 1992). ### Asset process and Black-Scholes-Merton (BSM) framework - Asset dynamics under risk-neutral measure Q: - dA(t) = r A(t) dt + A(t) dWt σA (equation (A1.1)) - Discrete/lognormal analog (A1.2) and physical probability distribution for horizon T–t given in (A1.3). - Relationship between assets, equity, and debt: - A(t) = E(t) + D(t). - Equity is the residual claim; equity holders effectively hold a call option on assets with strike equal to the default barrier. ### Distance to Default (DD) and probability of default (PD) - Firm leverage defined as B/A(t) (repayable face value of outstanding debt B divided by asset value). - Expected (physical) probability of default at time t expressed in (A1.4) and (A1.5). - Distance to default (DD) measure (equation (A1.6)): - d = [ln(At/B) + (r + 1/2 σA^2)(T−t)] / [σA sqrt(T−t)] with sign and formulation as in (A1.6). - Survival probability and relation to DD given in (A1.7). ### Equity and implicit option representations - Equity as a call option (risk-neutral expectation of termination payoff) (A1.8). - Closed-form value of equity as call option (A1.9). - Value of risky debt equals default-free debt minus present value of expected loss (A1.15). - Market-implied expected loss on liabilities computed as implicit put option (A1.16). ### Dividend adjustments and “lumpy dividend” treatment - Dividends S are modeled as “lumpy dividend” payments (footnote 80). - Discounted dividends up to horizon T represented in equation (A1.10): - 102 00 12 ()()( ) ... N rt trt trt t T SeS eSeS eS   = + + + . - Asset value after dividend payout: A(t) = Â(t) − S (A1.11). - Volatility relation with lumpy dividends: Âσ = σA (A1.12). - Generalized asset change: Â(t) − A(t) ≡ δA (A1.13). - Volatility adjustment specification (A1.14): - σ̂A = σA + γ δA (with γ calculated empirically). ### Implied asset volatility and system of equations - Itô-Döblin application yields equity diffusion expression (A1.17). - Assuming equity follows lognormal GBM (A1.18). - Matching conditions lead to expressions (A1.19) and (A1.20). - Implicit solution for asset volatility embedded in equity option formula presented in (A1.21). ### Decomposition of implicit put option into PD and LGD - Implicit put P(t) decomposed into PD and LGD components by re-arranging (A1.16) as shown in (A1.22). - Equation (A1.22) presents P(t) = PD · LGD decomposition algebraically, avoiding the need to assume a specific LGD. ### Risk-neutral density (state price density) and risk measures - Risk-neutral probability density (SPD) of A(t) is log-normal; functional form provided in (A1.23) with mean and variance details. - Risk measures (derivatives of expected default loss): - Δdef ≡ ∂Edef/∂A = Φ (equation (A1.24)). - Γdef ≡ ∂^2 Edef/∂A^2 = Φ (equation (A1.25)) with the exact functional forms as in the source. ### Practical estimation notes and limitations - Market-observable inputs: market value of equity and volatility of equity returns. - Equations (A1.9) and (A1.20) can be used to solve for implied asset value and asset return volatility. - The BSM-based system has empirical shortcomings; the main text presents an advanced CCA version addressing these issues. *Source: Appendix 1, Chapter 3 (IMF Working Paper), as provided in the supplied content.* ### Appendix 2. Estimation of the Empirical State Price Density (SPD) ### _wp1354 - Appendix 2. Estimation of the Empirical State Price Density (SPD) ### Extraction of the SPD via butterfly spreads (Breeden and Litzenberger, 1978) - Core idea: - Arrow-Debreu prices can be replicated via the butterfly spread on European call options, selling two calls at strike K and buying two calls at adjacent strikes K−ΔK and K+ΔK with stepsize ΔK. - The butterfly payoff Zγ of 1/ΔK of such spreads at time t (time to maturity T−t) is defined in equation (A2.1). - Key relations and limits: - Definitions: u1 and u2 as in (A2.2) and (A2.3) where C(·) denotes the price of a European call option. - As ΔK → 0 the butterfly position value becomes an Arrow-Debreu security paying 1 if AT = K (equation (A2.4)). - The position value tends to the second derivative of the call pricing function with respect to the strike when ΔK → 0 and C is twice differentiable (equation (A2.5)). - Combining these results yields the standard Breeden-Litzenberger relation: - f*(AT) = e^{r(T−t)} ∂^2 C(·)/∂K^2 evaluated at K = AT (equations (A2.6) and (A2.7)). - Assumptions and requirements: - No-arbitrage conditions; risk-free rate tr. - No assumptions on underlying asset dynamics or investor preferences (risk-neutrality with respect to the underlying only). - Markets must be perfect: no transaction costs or sales restrictions; agents can borrow and lend at the risk-free rate. ### Estimation of the GEV shape parameter using the Linear Combination of Ratios of Spacings (LRS) method (Appendix 3) - Purpose: - Use LRS as a natural estimator for the GEV shape parameter ξ̂ as initial value for maximum likelihood, given tail-dependent existence of raw moments. - LRS estimator specification: - ξ̂ = (1/4) ∑_{i=1}^n ln(v_{n,i}^c) − (1/4) ∑_{i=1}^n ln(4) (equation (A3.1)) — as presented in the source expression. - v_{n,i}^c defined via order statistics and quantiles per (A3.2). - Auxiliary constant a_c defined by a_c = ln(1 − a)^{-1} in (A3.3). - Asymptotic/statistical relation: - Approximation linking empirical order statistics and GEV parameters given in (A3.4). - Reference: - For more information on LRS, see Jobst (2007) (as cited in the source). ### Derivation of implied asset volatility using Moody’s KMV model under risk neutrality (Appendix 4) - Context: - Moody’s KMV estimated default frequency (EDF) under the physical measure is given by a Merton-style expression (A4.1). - For Systemic CCA expected-loss estimation under risk neutrality, the KMV expression must be adjusted to account for the market price of risk. - Risk-neutral re-expression: - EDF relation under risk-neutral measure is given in (A4.2) with parameters including correlation ρ_{AM}^{MKMV}, asset return r_A^{MKMV}, and the Market Sharpe Ratio (M μ − r)/σ_M as described in the text. - For T−t = 1 the re-arranged equation is shown in (A4.3), which leads to a quadratic in the adjusted asset volatility σ_A^*. - Recovery of adjusted asset volatility: - Quadratic solution structure indicated (quadratic formula sketch) and explicit formula for σ_A^* given in (A4.4). - From adjusted to actual asset volatility: - The actual asset volatility σ_A as a risk-neutral model input is obtained by solving the optimization constraint in (A4.5) over a rolling estimation window of τ days to match the intertemporal error-correction condition (A4.6). - Intertemporal error-correction specification linking KMV-reported volatilities: γ_t = (σ_{A,t}^{MKMV} + σ_{A,t+τ}^{MKMV}) / ... as per (A4.7) — text provides the structure of the intertemporal error correction over the estimation window. - Note: - The general asset volatility under risk-neutrality would need further refinement depending on the relevant option pricing model and its implications for asset volatility (footnote in source). ### Comparison of systemic risk measures versus Systemic CCA (Appendix 5) - Objective of systemic risk measures: - Determine contribution of individual financial institutions to systemic risk, capture spillover/contagion, and evaluate potential internalization via taxes, surcharges, or insurance premiums. - Methods compared: - CoVaR (Adrian and Brunnermeier, 2008): - Defines CoVaR as the VaR of the sector conditional on an institution being in distress. - Marginal contribution = CoVaR − system VaR. - Uses quantile regression; non-structural and motivated by bank-specific VaR tail risks. - Systemic Expected Shortfall (SES) and Marginal Expected Shortfall (MES) (Acharya and others): - MES = historical expected losses conditional on breach of a high systemic-risk threshold. - SES adjusts MES by firm-specific leverage/capitalization. - MES measures average, linear, bivariate dependence; does not consider interactions among subsets of banks and implicitly assumes sector undercapitalization. - Distress Insurance Premium (DIP) (Huang and others): - Hybrid approach combining equity and CDS information. - DIP = insurance cost to protect against distressed losses; summarizes market-perceived risk via expected default risk from CDS and correlations from equity returns. - Correlations from equity returns and default probabilities from CDS may reflect guarantees and do not isolate market-implied linkages fully. - Systemic Contingent Claims Approach (Systemic CCA): - Uses CCA for individual institutions and a multivariate distribution allowing nonlinear dependence to derive market-implied systemic risk from joint expected losses (and contingent liabilities). - Inputs: market data (equity and equity option prices) and capital structure information. - Produces a multivariate conditional tail expectation (CTE) measure based on Expected Shortfall (ES). - Advantages and methodological contrasts: - Tail risk representation: - ES is an improvement over VaR and is coherent, but conditioning ES on the most severe outcomes across banks can ignore the influence of the incidence of joint expected losses below the ES threshold. - Parametric SES/MES specifications based on quarterly data can be insensitive to rapidly changing market valuations and subject to parameter uncertainty. - Specification completeness and interpretability: - SES commingles leverage with non-specified default risk components (debt maturity, asset volatility), complicating identification of drivers and mitigants. - Systemic CCA adjusts for leverage and other structural determinants within a model-based framework. - Dependence structure: - DIP uses correlation from equity returns and default probabilities from CDS; this mixes signals that may be influenced by guarantees and captures only average linear bivariate dependence, not state-dependent or nonlinear dependence. - Systemic CCA models nonlinear, multivariate dependence allowing marginal contribution assessment at any confidence level and point in time without re-estimation. - Key claims about Systemic CCA: - Combines individual expected losses into joint expected losses, generating a conditional, non-linear systemic-risk sensitivity metric. - Quantifies marginal contribution of an individual firm while accounting for time-varying market-valued balance-sheet structures. - Allows valuation of systemic risk charges using current market conditions rather than historical experience. - Multivariate density estimation enables determination of marginal contributions across the entire distribution (not limited to specific percentiles). - Illustration details (Figure A1): - Stylized marginal rate of substitution (MRS) between individual contributions to systemic risk and joint impact of default risk is illustrated via bivariate kernel density (Chart 1) and contour plot (Chart 2). - Chart specifics: - Sample period: 482 observations of individual put option values of sample banks in a small European country with an outsized financial sector relative to GDP. - The bivariate distribution axes: y-axis = average contingent liabilities (alpha-value * implicit put option) in percent of GDP (as of end-2008); x-axis = total systemic risk from contingent liabilities in percent of GDP. - Kernel density estimation with Epanechnikov (1969) kernel function and linear binning, using an empirically derived bandwidth. - Conceptual contrasts: - CoVaR corresponds to the average systemic risk at a specific percentile (perpendicular red plane / red dashed line). - MES/SES covers density in a rectangle (right-side region defined by ES threshold). - Systemic CCA covers the entire surface/contour via multivariate density estimation, enabling marginal contributions across all confidence levels and times. *Source: _wp1354 - Appendix 2. Estimation of the Empirical State Price Density (SPD)* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp1354.pdf_