## wp1849

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### Purpose and proposed framework
- Aim: Provide an easily implementable and robust method that incorporates individual entity stress tests and losses from systemic risk effects (SE losses) into macroprudential stress tests.
- Core idea: An “encompassing” method combines stress tests on individual entities with an empirical reduced-form approach to quantify losses from systemic effects under adverse macroeconomic scenarios.
- Role of SE losses: SE losses are defined as losses suffered by financial entities conditional on macroeconomic adverse scenarios and the distress of other entities in the system; they are added to microprudential stress-test losses to obtain total losses in a systemic event.
- Reduced-form inference: “Reduced form” here infers interconnectedness from market data without explicitly modeling the generating mechanisms.

### Methodology and technical approach
- System representation:
  - Characterize the financial system as a portfolio of interconnected entities; typify entities’ asset values and their association via a multivariate density.
- Recommended estimator:
  - Consistent Information Multivariate Density Optimizing (CIMDO) recommended (see Section VI.B and Appendix III).
  - CIMDO is based on the minimum cross-entropy approach: a prior multivariate distribution is updated via an optimization procedure using empirical constraints (individual banks’ probabilities of distress, PoDs) to recover a posterior (the CIMDO density).
- Conditional loss valuation:
  - An asset valuation model coupled with the multivariate density quantifies expected losses suffered by specific entities conditional on other entities falling into distress; these conditional losses constitute SE losses.
  - This approach infers implied direct and indirect channels of contagion without structural contagion assumptions and is stochastic.
- Integration with stress-testing:
  - After a microprudential stress test identifies entities in distress from first-round shocks, the proposed approach identifies entities that would fall into distress after incurring SE losses (second-round effects).

### Advantages and scope
- Data flexibility:
  - Can be estimated with publicly available data, including market-based or supervisory information on PoDs, enabling implementation across a broad set of countries.
- Market-consistent interconnectedness:
  - When PoDs use market data, CIMDO embeds market perceptions of direct and indirect asset-value interconnectedness structures and captures nonlinear increases in losses observed in crises.
- Multivariate dependence:
  - CIMDO infers the entire multivariate density—including linear and nonlinear dependence (the copula function)—rather than relying solely on pairwise correlations.
- Inclusion of nonbank intermediaries:
  - Framework permits integration of insurance companies, pension funds, investment funds, and hedge funds into systemic risk analysis.
- Practicality versus simulation models:
  - Unlike simulated network or agent-based models, the empirical reduced-form approach does not require highly granular data or explicit behavioral modeling, easing implementation and broad applicability.

### Limitations and caveats
- Reduced-form nature:
  - Empirical reduced-form models capture effects of agents’ behavior but do not identify specific behavioral channels of contagion.
- Market-data issues:
  - Metrics derived from market data may be affected by illiquidity or market imperfections.
- Structural detail trade-off:
  - Although the approach infers interconnectedness structures consistent with observed PoDs, it does not model the micro-level mechanisms that generate those structures.

### Implementation and loss definitions (MicroST and SE)
- Microprudential stress test valuation:
  - VA_MicroST = E(VA | adv)
  - Loss_micro(A) = E(VA) − E(VA | adv)
- Systemic-effect (SE) loss definitions:
  - Loss_SE(A | S) = E(VA | adv) − E(VA | adv ∩ S)
  - Loss_TS(A | S) = Loss_micro(A) + Loss_SE(A | S) = E(VA) − E(VA | adv ∩ S)
- Reasons why MicroST misses SE losses:
  - Do not capture non linearities in mappings from macro scenarios to risk parameters.
  - Omit contagion mechanisms across entities and markets.
  - Conditioning on specific entity defaults (adv ∩ S) can widen conditional losses when the adverse scenario is stochastic.
- Implementation steps under a deterministic scenario:
  - Estimate scenario-specific risk parameters (probabilities of default, loss given default, exposures at default) for each FI.
  - Use parameters to estimate MicroST losses and identify entities that may experience capital shortfalls after second-round effects.
  - Quantify losses from contagion; identify whether FIs would survive additional SE losses; calculate each connecting entity’s contribution to contagion losses decomposed into likelihood and intensity.

### Quantifying SE losses and vulnerability index
- SE loss when S = default of bank Aj:
  - Loss_SE(Ai | Aj) = E(V_Ai | adv) − E(V_Ai | Aj ∩ adv)
  - Loss_TS(Ai | Aj) = Loss_micro(Ai) + Loss_SE(Ai | Aj) = E(V_Ai) − E(V_Ai | Aj ∩ adv)
- Vulnerability index:
  - V(Ai | Aj) = Loss_TS(Ai | Aj) / TA(Ai)
  - TA(Ai) denotes total assets of Ai. Higher V(Ai | Aj) indicates stronger impact of Aj’s default on Ai’s losses.
- Note:
  - High conditional SE loss of Ai given Aj’s default does not necessarily imply a strong direct A–B connection; contagion paths may involve other intermediaries.

### Decomposing SE losses (probabilities, intensities, contributions)
- Law of total expectation decomposition (four-FI example with A, B, C, D) when B defaults:
  - {B} = {B∩C∩D, B∩C̅∩D, B∩C∩D̅, B∩C̅∩D̅}
  - Loss_SE(A | B) =
    P(B∩C∩D | B) Loss_SE(A | B∩C∩D)
    + P(B∩C̅∩D | B) Loss_SE(A | B∩C̅∩D)
    + P(B∩C∩D̅ | B) Loss_SE(A | B∩C∩D̅)
    + P(B∩C̅∩D̅ | B) Loss_SE(A | B∩C̅∩D̅)
- Interpretation:
  - SE loss of A given B equals the probability-weighted average of loss intensities across all configurations of other FIs’ survival/default states given B defaults.
  - Two drivers: probability of a specific joint-default event and expected loss suffered by A under that event.
- Generalized formula for N firms:
  - Loss_SE(Ai | Aj) = Σ P(Dji(k1,...,kl) | Aj) Loss_SE(Ai | Dji(k1,...,kl))
  - Where Dji(k1,...,kl) denotes event where Aj and {Ak1,...,Akl} default and all other FIs (except Ai) do not default.

### Contribution and intensity indexes (definitions preserved)
- Contribution index:
  - Co(Dji(k1,...,kl) | Aj) =
    P(Dji(k1,...,kl) | Aj) · Loss_SE(Ai | Dji(k1,...,kl)) / Loss_SE(Ai | Aj)
  - Interpretation:
    - Co is between 0 and 1.
    - Identifies the “connecting” entities that contribute most to SE losses from Aj to Ai.
    - Large Co can arise from high conditional probability or large conditional loss intensity.
    - Multivariate distribution cannot identify causality; “due” is not causal.
- Intensity ratio:
  - In(Dji(k1,...,kl) | Aj) = Loss_SE(Ai | Dji(k1,...,kl)) / Loss_SE(Ai | Aj)
  - Interpretation:
    - Ratio equals SE loss assuming realization of event D divided by SE loss given Aj’s default.
    - Range: between zero and infinity.

### V. Quantifying SE losses — portfolio representation and requirements
- Key distinction:
  - Financial system represented as a portfolio of financial entities allowing SE losses to be defined as losses suffered by specific entities conditional on distress of other entities.
- Requirements for quantification:
  - Estimation of conditional and unconditional entities’ asset expected values.
  - Use of an asset valuation model (structural approach).
  - Estimation of multivariate densities (CIMDO).

### Asset Valuation Model (structural multivariate extension)
- Basis:
  - Structural approach to corporate default (Merton (1974)); firm’s asset value follows a log-normal process; default triggered by asset value dropping below a default threshold modeled as a function of leverage.
  - Merton’s univariate approach extended to multivariate case; model presented for two FIs and generalized to N FIs.
- Unconditional valuation:
  - E0(VA,t)= ∫∫ VA,t(x,y) p(x,y) dx dy
  - VA,t(x,y)= Eq t(x,y) + Dt(x,y)
  - Debt valuation:
    - If A defaults (x > Xx d): Dt(x,y) = (RR) DT e−r(T−t)
    - If A does not default (x < Xx d): Dt(x,y) = DT e−r(T−t)
  - Expected value split:
    - E0(VA,t)= Eq0 ∫∫ e−x p(x,y) dx dy + P(A)(RR)DT e−r(T−t) + P(A̅)DT e−r(T−t)
  - P(A) is the marginal probability of default of A.
- Conditional valuation (conditioning on default of B):
  - E0(VA,t | B) = 1/P(B) ∫∫ VA,t(x,y) p(x,y) I(y > Xy d) dx dy
  - Expanded:
    - E0(VA,t | B)= 1/P(B) ∫∫ Eq t(x,y) p(x,y) I(y > Xy d) dx dy
      + P(A ∩ B)/P(B) (RR)DT e−r(T−t)
      + P(A̅ ∩ B)/P(B) DT e−r(T−t)
- SE losses under event S:
  - Loss_SE(A | S) = E(VA | adv) − E(VA | adv ∩ S)

### CIMDO method (Appendix I summary)
- Purpose:
  - Recover multivariate distributions of FIs’ implied asset values without imposing restrictive parametric assumptions.
- Challenges:
  - Lack of data for tail events; unobservability of dependence structure; inadequacy of normality/fixed-dependence assumptions.
- Method:
  - CIMDO = Consistent Information Multivariate Density Optimization based on Kullback cross-entropy.
  - Update prior density q (calibrated as multivariate t-distribution using FIs equity returns) with empirical constraints (individual PoDs) to obtain posterior p, minimizing cross-entropy subject to additivity and marginal constraints.
- Advantages:
  - Infers dependence structures consistent with observed individual PoDs at specific times; dependence structures update as empirical PoDs change; can be estimated using market or supervisory information.
  - CIMDO-inferred densities forecast better than parametric distributions calibrated with same information set (using PIT criterion).
- PoD inputs:
  - Distress includes default, debt restructuring, government intervention, recapitalization, downgrades, and other events causing significant asset value decreases.
  - Estimation approaches: Merton-type models; CDS-spreads and bond spreads (risk-neutral to subjective probabilities conversion); supervisory-defined buffer thresholds and simulated loss distributions; Value at Risk for investment funds.
  - Input choice implications: market-based inputs embed market-price channels; supervisory inputs may omit market-price channels.

### Application: U.S. banking system case study (simplified 4-bank system, 2008)
- System: Citibank (C), Lehman Brothers (LB), Wells Fargo (WFC), Morgan Stanley (MS).
- Objective: compute expected losses for C, WFC, and MS assuming LB default, moving from microprudential outputs to macroprudential SE estimates.
- SE losses (as percent of each bank’s total assets) if Lehman defaulted:
  - Citibank (C): 6.7 percent
  - Wells Fargo (WFC): 5.6 percent
  - Morgan Stanley (MS): 10.0 percent
- TARP capital injections (Date: 11/13/2008; amounts in millions of USD) and comparisons:
  - Citibank (C):
    - Capital Purchase Program 25,000
    - Targeted Investment Program 20,000
    - Total Injection 45,000
    - Total Injection / SE Losses assuming LB default / Total Assets = 2.3
    - Total Assets = 6.7
  - Wells Fargo (WFC):
    - Capital Purchase Program 25,000
    - Targeted Investment Program 0
    - Total Injection 25,000
    - Total Injection / SE Losses assuming LB default / Total Assets = 4.0
    - Total Assets = 5.6
  - Morgan Stanley (MS):
    - Capital Purchase Program 10,000
    - Targeted Investment Program 0
    - Total Injection 10,000
    - Total Injection / SE Losses assuming LB default / Total Assets = 1.0
    - Total Assets = 10.0
- Interpretation:
  - SE loss estimates were of a similar order of magnitude to capital injections but larger than actual injections, possibly because market-expected losses were estimated right after Lehman’s default (September 2008), before recapitalization took effect in November 2008.
  - Possible inference: U.S. government recapitalization program was effective in containing default cascades and systemic risk losses, reducing realized needs relative to initial market expectations.

### Decomposition results (Citibank conditional on Lehman default)
- Decomposition components (for defaulting sets; values preserved exactly):
  - Pr(C | LB): 66.7; 27.5; 1.3; 3.7
  - In(C | LB): 0.52; 1.76; 2.02; 3.89
  - Co(C | LB): 34.5; 48.4; 2.7; 14.5
- Key insights:
  - Largest contribution to Citibank’s SE losses given LB default is from conditional loss when LB and MS default but not WFC: Co = 48.4 percent; Pr ≈ 27.5 percent; In = 1.76.
  - Defaulting set LB and WFC but not MS has higher loss intensity (In = 2.02) but very low probability (Pr ≈ 1.3 percent), yielding Co = 2.7 percent.
  - Co equals Pr multiplied by In for each defaulting set; sum of Co equals 100 percent.
- Conditional loss scaling (Table 3 values, in Millions of USD):
  - 41,049.24
  - 191,397.00
  - 303,513.05
  - 460,935.80

### Expected and book asset values — consistency checks (values preserved)
- Expected asset values (Table 4a, in Millions USD):
  - Sep-08:
    - C: 2,052,764
    - WFC: 650,171
    - MS: 982,022
  - Dec-08:
    - C: 1,896,820
    - WFC: 1,319,273
    - MS: 653,108
- Book value of assets (Table 4b, in Millions USD; company reports):
  - Sep-08:
    - C: 2,050,131
    - WFC: 622,361
    - MS: 987,403
  - Dec-08:
    - C: 1,938,470
    - WFC: 1,309,639
    - MS: 658,812
- Observation: Expected asset values are consistent with reported book values.

### Implications for calibration of capital buffers and policy uses
- Basel III context:
  - Capital Conservation Buffer (CCoB) size referenced: "2.5 percent of RWA on top of the minimum capital requirement outside periods of stress." The buffer can be drawn down in stress periods.
  - Countercyclical Capital Buffer (CCyB) can vary between zero and 2.5 percent of RWA; BCBS buffer guide uses the credit-to-GDP gap with thresholds 2 percent (lower) and 10 percent (upper) mapping to a 2.5 percent CCyB when breached.
  - Systemically Important Bank (SIB) capital surcharge applied via BCBS methodologies; buckets range from 1 to 3.5 percent.
- Use of stress tests:
  - Macroprudential stress tests useful to calibrate CCyB and SIB surcharges; country examples include United Kingdom and United States (proposals for bank-specific SCB equal to maximum decline of tier 1 under severe adverse scenario).
- Role of SE loss estimation:
  - Estimating SE losses (including augmentation for severity of financial imbalances) can inform CCyB calibration to ensure buffers allow banks to withstand amplified losses at cycle peaks.
- Framework proposition advantages:
  - Combines existing microprudential stress tests with reduced-form SE estimation; performs with publicly available data; embeds market perceptions of indirect interconnectedness; avoids explicit behavioral reaction modeling; stochastic structure enables event probabilities and loss intensities; multivariate dimension facilitates integration of nonbanks; cost-efficient and relatively light on data.
- Extensions and further work:
  - Hiebert and others (2018): augment framework to estimate amplification conditional on severity of financial imbalances for CCyB calibration.
  - Espinoza and others (2018): combine a general equilibrium model with the reduced-form approach to incorporate systemic risk endogeneity and calibrate nonlinear amplification mechanisms.

*Source: wp1849*

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

### wp1849 - References .............................................................................................................

### Purpose and proposed framework
- Aim: Provide an easily implementable and robust method that incorporates individual entity stress tests and losses from systemic risk effects (SE losses) into macroprudential stress tests.
- Core idea: An “encompassing” method combines stress tests on individual entities with an empirical reduced-form approach to quantify losses from systemic effects under adverse macroeconomic scenarios.
- Role of SE losses: SE losses are defined as losses suffered by financial entities conditional on macroeconomic adverse scenarios and the distress of other entities in the system; they are added to microprudential stress-test losses to obtain total losses in a systemic event. Footnote 2 clarifies that “reduced form” here infers interconnectedness from market data without explicitly modeling the generating mechanisms.

### Methodology and technical approach
- Asset-value multivariate density: The framework characterizes a financial system as a portfolio of interconnected entities and typifies the entities’ asset values and their association via a multivariate density.
- Recommended estimator: Consistent Information Multivariate Density Optimizing (CIMDO) is recommended (see Section VI.B and Appendix III).
  - CIMDO is based on the minimum cross-entropy approach: a prior multivariate distribution is updated via an optimization procedure using empirical constraints (individual banks’ probabilities of distress, PoDs) to recover a posterior (the CIMDO density). Footnote 3 explains this process and cites Segoviano (2006) and Segoviano and Espinoza (2017).
- Conditional loss valuation: An asset valuation model coupled with the multivariate density quantifies expected losses suffered by specific entities conditional on other entities falling into distress; these conditional losses constitute SE losses. Footnote 6 clarifies that this approach infers implied direct and indirect channels of contagion without structural contagion assumptions and is stochastic.
- Integration with stress-testing: After a microprudential stress test identifies entities in distress from first-round shocks, the proposed approach identifies entities that would fall into distress after incurring SE losses (second-round effects).

### Advantages and scope
- Data flexibility: Can be estimated with publicly available data, including market-based or supervisory information on PoDs, enabling implementation across a broad set of countries.
- Market-consistent interconnectedness: When PoDs use market data, CIMDO embeds market perceptions of direct and indirect asset-value interconnectedness structures and captures nonlinear increases in losses observed in crises.
- Multivariate dependence: CIMDO infers the entire multivariate density—including linear and nonlinear dependence (the copula function)—rather than relying solely on pairwise correlations.
- Inclusion of nonbank intermediaries: The multivariate framework permits the integration of insurance companies, pension funds, investment funds, and hedge funds into systemic risk analysis (see footnote 7 and Cortes and others (2018) for an extension).
- Practicality versus simulation models: Unlike simulated network or agent-based models, the proposed empirical reduced-form approach does not require highly granular data or explicit behavioral modeling, easing implementation and broad applicability.

### Limitations and caveats
- Reduced-form nature: Empirical reduced-form models capture effects of agents’ behavior but do not identify specific behavioral channels of contagion.
- Market-data issues: Metrics derived from market data may be affected by illiquidity or market imperfections.
- Structural detail trade-off: Although the approach infers interconnectedness structures consistent with observed PoDs, it does not model the micro-level mechanisms that generate those structures (contrast with network/ABM approaches discussed and limitations noted in footnote 4 and related text).

### Implementation notes and references within the paper
- Relevant sections and appendices: Section VI.B and Appendix III describe the CIMDO approach; Appendix IV contrasts the SE-loss approach with network models; Appendices I–III generalize conditional valuation and SE concepts. The paper’s structure proceeds from theoretical foundations (Section II) to implementation and applications.
- Practical interpretation: SE losses quantify conditional expected losses under distress events and can be combined with microprudential stress-test outputs to obtain second-round system-wide loss estimates.

*Source: wp1849 - References .............................................................................................................*

### Section III formalizes the encompassing method that we propose and explains how it can be

### wp1849 - Section III formalizes the encompassing method that we propose and explains how it can be implemented

### Systemic risk: nature and modelling approaches
- Systemic effects (SE) arise from generalized shocks, contagion through direct contractual interconnectedness, and indirect interconnectedness (common exposures, asset fire sales, information asymmetries). References in the source include Bernanke and others 1999; Kiyotaki and Moore 1997; Adrian and Shin 2014; Allen and Gale 2000; Freixas and others 2000; Eisenberg and Noe 2001; Bhattacharya et al. 2007; Bhattacharya and Gale 1987; Aspachs et al. 2007; Lorenzoni 2008; Jacklin and Bhattacharya 1988; Khandani and Lo 2011.
- Interconnectedness structures are complex and unstable during distress, producing nonlinear increases in magnitude and speed of loss propagation observed in crises.
- Two main empirical approaches:
  - Simulated models (network models, Cifuentes, Ferrucci, and Shin 2005; Eisenberg and Noe 2001; Alessandri and others 2009; Aikman and others 2009; Tressel 2010) trace amplification mechanisms but typically focus on specific channels, rely on strong structural assumptions (elasticities of asset prices to fire sales, behavioral reaction functions), require granular data, and often omit simultaneous effects. Limitations have contributed to difficulty generating realistic contagion loss magnitudes observed in the global financial crisis (Elsinger and others 2013). Improvements noted (Cont and Schaanning 2016). Appendix IV summarizes Eisenberg and Noe (2001) and Cifuentes and others (2005).
  - Agent-based models (ABM) introduce heterogeneous agents and behavioral responses; increase computational complexity and data needs as features are added.
  - Empirical reduced-form models infer interconnectedness from market co-movements; they embed direct and indirect contagion reflected in market data and are suitable for high-frequency monitoring but do not isolate specific amplification mechanisms. Examples: Diebold and Yilmaz (2009), Segoviano and Goodhart (2009), Adrian and Brunnermeier (2016), Acharya and others (2017).

### The encompassing method: purpose and core features
- Objective: develop an operational macroprudential stress test that combines microprudential stress tests and reduced-form, market-based systemic risk models.
- Key features:
  - Leverages existing microprudential stress tests (bottom-up or top-down) and supervisory or publicly available data focusing on individual entities’ fundamentals.
  - Estimates SE losses via a reduced-form approach that incorporates market perceptions of financial-system interconnectedness; SE loss estimates embed realistic market reactions, are computationally simple, and relatively light on data requirements.
  - Can be used alongside alternative simulated models to improve calibration; cost-efficiency allows parallel running for enhanced policymaker insight.

### Implementation and loss definitions (MicroST and SE)
- Microprudential stress test valuation:
  - VA_MicroST = E(VA | adv)  (equation (1))
  - MicroST loss of bank A:
    - Loss_micro(A) = E(VA) − E(VA | adv)  (equation (2))
- Systemic-effect (SE) loss definitions:
  - SE loss of bank A conditional on financial contagion event S:
    - Loss_SE(A | S) = E(VA | adv) − E(VA | adv ∩ S)  (equation (3))
  - Total loss under a systemic event TS (realization of S in the stressed macro scenario):
    - Loss_TS(A | S) = Loss_micro(A) + Loss_SE(A | S) = E(VA) − E(VA | adv ∩ S)  (equation (4))
- Reasons microprudential stress tests typically miss SE losses:
  - Do not adequately capture non linearities affecting mapping from macro scenarios to risk parameters (probabilities of default, loss given default, etc.) and portfolio diversification effects.
  - Omit contagion mechanisms across entities and markets.
  - When adverse scenario is stochastic, knowledge of specific entity defaults narrows the set of consistent adverse scenarios (adv ∩ S), making conditional losses potentially larger.
- Implementation steps under a deterministic scenario (per Figure 2 description):
  - Estimate scenario-specific risk parameters (probabilities of default, loss given default, exposures at default) for each FI analyzed; integration of banks and nonbanks is straightforward.
  - Use parameters to estimate MicroST losses and to identify entities that may experience capital shortfalls after second-round effects.
  - The framework quantifies losses from contagion; identifies whether FIs would survive additional SE losses given defaults of specific entities; and calculates each “connecting” entity’s contribution to contagion losses decomposed into likelihood and intensity.

### Quantifying SE losses and vulnerability index
- When S is the distress/default of a specific bank Aj, for institution Ai:
  - Loss_SE(Ai | Aj) = E(V_Ai | adv) − E(V_Ai | Aj ∩ adv)  (equation (5))
  - Loss_TS(Ai | Aj) = Loss_micro(Ai) + Loss_SE(Ai | Aj) = E(V_Ai) − E(V_Ai | Aj ∩ adv)  (equation (6))
- Vulnerability index of Ai to Aj’s default:
  - V(Ai | Aj) = Loss_TS(Ai | Aj) / TA(Ai)  (equation (7))
  - TA(Ai) denotes total assets of Ai. Higher V(Ai | Aj) indicates stronger impact of Aj’s default on Ai’s losses.
- Notes:
  - High conditional SE loss of Ai given Aj’s default does not necessarily imply a strong direct A–B connection; contagion paths may involve other strongly connected intermediaries.

### Decomposing SE losses (example and interpretation)
- Decomposition clarifies probabilities and intensities of defaulting sets under the conditioning that a particular bank defaults.
- Four-FI example (A, B, C, D): partition when B defaults:
  - {B} = {B∩C∩D, B∩C̅∩D, B∩C∩D̅, B∩C̅∩D̅}  (equation (8))
- Law of total expectation decomposition of Loss_SE(A | B):
  - Loss_SE(A | B) =
    P(B∩C∩D | B) Loss_SE(A | B∩C∩D)
    + P(B∩C̅∩D | B) Loss_SE(A | B∩C̅∩D)
    + P(B∩C∩D̅ | B) Loss_SE(A | B∩C∩D̅)
    + P(B∩C̅∩D̅ | B) Loss_SE(A | B∩C̅∩D̅)  (equation (9))
- Interpretation:
  - SE loss of A given B equals the probability-weighted average of loss intensities across all configurations of other FIs’ survival/default states given B defaults.
  - Two drivers matter: probability of a specific joint-default event and expected loss suffered by A under that event.
- Extension note: SE losses can be conditioned on the severity of financial imbalances (leverage, mispricing of risk, liquidity and maturity mismatches). For a given macro shock, SE losses increase with the severity of imbalances (reference to Hiebert et al (2018)).

### Generalized SE loss formula for N firms
- For network of N firms {A1, ..., AN}, denote Dji(k1,...,kl) as event where Aj and {Ak1,...,Akl} default and all other FIs (except Ai) do not default.
- SE loss expression:
  - Loss_SE(Ai | Aj) = E(V_Ai) − E(V_Ai | Aj)
    = E(V_Ai) − Σ P(Dji(k1,...,kl) | Aj) E(V_Ai | Dji(k1,...,kl))
    = Σ P(Dji(k1,...,kl) | Aj) [E(V_Ai) − E(V_Ai | Dji(k1,...,kl))]
    = Σ P(Dji(k1,...,kl) | Aj) Loss_SE(Ai | Dji(k1,...,kl))  (equation (10))
- Interpretation:
  - Loss_SE(Ai | Aj) is the weighted average across all possible combinations of other banks defaulting, where weights are conditional probabilities P(Dji(...) | Aj) and outcomes are the SE losses induced by each default set.

*Source: wp1849 - Section III formalizes the encompassing method that we propose and explains how it can be implemented.*

### Appendix II extends this formula for any conditioning event S (made of the default of k FIs

### wp1849 - Appendix II extends this formula for any conditioning event S (made of the default of k FIs and the non-default of N-k-1 FIs)

### Contribution of connecting entities to SE losses
- Definition (equation as given):

  퐶표(퐷푗푖(푘1,...,푘𝑙)|퐴푗)=
  푃(퐷푗푖(푘1,...,푘𝑙)|퐴푗)
  퐿표푠푠푆퐸(퐴푖|퐷푗푖(푘1,...,푘𝑙))
  퐿표푠푠푆퐸(퐴푖|퐴푗)  (11)

- Interpretation and properties:
  - The index 퐶표(퐷푗푖(푘1,...,푘𝑙)|퐴푗) is the contribution to the conditional loss experienced by the set 퐷푗푖(푘1,...,푘𝑙) due to the default of 퐴푗.
  - As a contribution, Co is between 0 and 1.
  - This measure identifies the “connecting” entities that contribute most to the SE losses from 퐴푗 to 퐴푖.
  - The contribution can be large because:
    - the connecting banks are likely to default (high probability term), or
    - the intensity of the losses experienced by 퐴푖 are large given a default of the connecting FIs (large conditional loss term).
  - Co represents the intensity of losses for the set of defaults 퐷푗푖(푘1,...,푘𝑙), weighted by the probability of occurrence of such event.
  - Footnote: Since the multivariate distribution framework cannot identify causality, the word ‘‘due’’ should not be interpreted as suggesting a causal link.

### Intensity of conditional losses (ratio)
- Definition (equation as given):

  퐼푛(퐷푗푖(푘1,...,푘𝑙)|퐴푗)=
  퐿표푠푠푆퐸(퐴푖|퐷푗푖(푘1,...,푘𝑙))
  퐿표푠푠푆퐸(퐴푖|퐴푗)  (12)

- Interpretation:
  - This ratio equals the SE loss assuming realization of event 퐷푗푖(푘1,...,푘𝑙) divided by the SE loss given 퐴푗’s default.
  - Indicates the relative weight of the conditional losses induced by a given set of defaults over the SE loss induced by the default of bank 퐴푗.
  - Larger when losses due to 퐷푗푖(푘1,...,푘𝑙) are higher.
  - Range: between zero and infinity.

### V. Quantifying SE losses — portfolio representation and requirements
- Key distinction:
  - Financial system represented as a portfolio of financial entities allowing SE losses to be defined as losses suffered by specific entities conditional on distress of other entities.
- Requirements for quantification:
  - Estimation of conditional and unconditional entities’ asset expected values.
  - Use of an asset valuation model (presented in section VI.A).
  - Estimation of multivariate densities (presented in section VI.B).

### A. Asset Valuation Model (structural approach, multivariate extension)
- Basis:
  - Structural approach to corporate default (Merton (1974)): firm's underlying asset value follows a log-normal process; default triggered by asset value dropping below a default threshold modeled as a function of leverage.
  - Merton’s univariate approach extended to multivariate case.
  - Model presented for a portfolio of two FIs (generalized to N FIs in Appendix I).
- Assumptions:
  - Multivariate distribution of asset returns of FIs A and B, 푝(푥,푦), has been estimated.
  - Default thresholds 푋푥푑 and 푋푦푑 for asset returns x and y are known.
  - Formulas valid with or without conditioning on an adverse macroeconomic scenario; to condition on adverse scenario replace 푝(푥,푦) by 푝(푥,푦|푎푑푣).

### Unconditional valuation
- Expected (unconditional) value of assets of firm A (equation as given):

  E0(VA,t)=
  ∫∫
  푉퐴,t(푥,푦)
  푝(푥,푦)
  푑푥푑푦       (13)

- Decomposition of asset value:

  푉퐴,t(푥,푦)=퐸푞푡(푥,푦)+퐷푡(푥,푦)       (14)

- Debt valuation depending on default state:
  - If A defaults (푥>푋푥푑): 퐷푡(푥,푦)=(푅푅)퐷푇 푒−푟(푇−푡)
  - If A does not default (푥<푋푥푑): 퐷푡(푥,푦)=퐷푇 푒−푟(푇−푡)
  - r is the discount rate and RR is recovery rate (one minus the loss given default).

- Expected value splitting between debt and equity (equation as given):

  퐸0(푉퐴,t)=퐸푞0
  ∫∫
  푒−푥
  푝(푥,푦)
  푑푥푑푦+푃(퐴)(푅푅)퐷푇 푒−푟(푇−푡)+푃(퐴̅)퐷푇 푒−푟(푇−푡)  (15)

- Note:
  - P(A) is the (marginal) probability of default of A, estimated by integrating p(x,y) over A's default zone.
  - 푃(퐴̅)=1−푃(퐴).

### Conditional valuation (conditioning on default of B)
- Conditional expected asset value of A given default of B (equations as given):

  퐸0(푉퐴,t|퐵)=
  1
  푃(퐵)
  퐸(푉퐴,t 1𝐵)=
  1
  푃(퐵)
  ∫∫
  푉퐴,t(푥,푦)
  푝(푥,푦)
  퐼(푦>푋푦푑)
  푑푥푑푦  (16)

  where 푋푦푑 is the threshold for B's asset return above which B is in default.

- Splitting debt and equity under conditioning (equation as given):

  퐸0(푉퐴,t|퐵)=
  1
  푃(퐵)
  ∫∫퐸푞푡(푥,푦)
  푝(푥,푦)
  퐼(푦>푋푦푑)
  푑푥푑푦 +
  1
  푃(퐵)
  ∫∫
  퐷푡(푥,푦)
  푝(푥,푦)
  퐼(푦>푋푦푑)
  푑푥푑푦   (17)

- Expanded conditional valuation (equation as given):

  퐸0(푉퐴,t|퐵)=
  1
  푃(퐵)
  ∫∫퐸푞0 푒−푥
  푝(푥,푦)
  퐼(푦>푋푦푑)
  푑푥푑푦 +
  푃(퐴∩퐵)
  푃(퐵)
  (푅푅)퐷푇 푒−푟(푇−푡)
  +
  푃(퐴̅∩퐵)
  푃(퐵)
  퐷푇 푒−푟(푇−푡)        (18)

### SE losses under event S
- SE losses for A under event S (equation as given):

  퐿표푠푠푆퐸(퐴|푆)=퐸(푉퐴|푎푑푣 )−퐸(푉퐴|푎푑푣 ∩푆)      (19)

- Use:
  - Difference between conditional and unconditional valuation used to quantify systemic risk amplification loss.

*Source: wp1849 - Appendix II extends this formula for any conditioning event S (made of the default of k FIs)*

### Appendix I provides the general formulas when the event S is any combination of k FIs

### wp1849 - Appendix I provides the general formulas when the event S is any combination of k FIs

### The CIMDO Method
- Purpose: Recover multivariate distributions of FIs' implied asset values without imposing restrictive parametric assumptions.
- Key challenges addressed:
  - Lack of data for tail events.
  - Unobservability of the dependence structure that defines joint distress.
  - Normality and fixed-dependence assumptions of standard multivariate Gaussian extensions are inadequate because financial asset returns exhibit heavier tails and time-varying interconnectedness.
- Methodology:
  - CIMDO = Consistent Information Multivariate Density Optimization, based on Kullback (1959) cross-entropy approach.
  - Uses observed information (entities’ equity returns and individual PoDs) to infer the unobserved dependence structure embedded in the multivariate density.
  - Recovers a posterior multivariate distribution (the CIMDO density) by updating a prior density q (calibrated as a multivariate t-distribution using FIs equity returns data) with empirical constraints (individual PoDs).
  - Optimization objective: find posterior p that remains as close as q as possible subject to additivity constraint (distribution sums to 1) and marginal constraints (consistency with observed PoDs).
- Advantages relative to parametric approaches:
  - Infers dependence structures consistent with observed individual PoDs at specific times, reducing risk of density misspecification.
  - Dependence structures are updated as empirical PoDs change, enabling timeliness and capturing nonlinear increases in interconnectedness during high volatility.
  - Can be estimated using market information or supervisory information; does not require highly granular supervisory data.
- Additional notes:
  - CIMDO-inferred densities forecast better than parametric distributions calibrated with the same information set (using an extension of the Probability Integral Transformation (PIT) criterion).
  - CIMDO dependence structures embody both linear and nonlinear distress dependence and are time-varying.
  - Individual PoDs used as inputs are exogenous and can be estimated from market data (e.g., CDS spreads, stock returns) or supervisory/balance-sheet methods.

### Probabilities of Distress (PoD) — Inputs
- Definitions and scope:
  - Distress events include default, debt restructuring, government intervention, recapitalization, credit agencies’ downgrades, and other events that produce significant decreases in asset values.
- Estimation approaches:
  - Market-based information:
    - Merton-type models (distress ≡ default per Merton 1974).
    - CDS-spreads and bond spreads (PoDs estimated from CDS or deduced from bond spreads; conversion from risk-neutral to subjective probabilities via market price of risk methods).
  - Supervisory information:
    - Use supervisory-defined capital-buffer thresholds and simulated portfolio loss distributions to estimate PoDs (Segoviano and Padilla 2006).
  - Investment funds:
    - PoD defined as probability that funds must liquidate assets to meet redemption demands; estimated via a Value at Risk approach (Cortes and others 2017).
- Input choice implications:
  - Market-based inputs embed market perceptions of indirect interconnectedness (exposures to common risk factors and market-price channels).
  - Supervisory-based inputs may capture indirect interconnectedness via common risk factors but might omit market-price channels.
- Implementation detail for the U.S. case study:
  - Used the probability of default obtained from CDS spreads for the day before Lehman Brothers defaulted.

### Application to the U.S. Banking System — Case Study Overview
- System: simplified 4-bank system in 2008 — Citibank (C), Lehman Brothers (LB), Wells Fargo (WFC), Morgan Stanley (MS).
- Objective: compute expected losses for C, WFC, and MS assuming LB default, moving from microprudential stress-test outputs to macroprudential system expected loss (SE) estimates.
- Important methodological point: SE losses estimated using equation 5 (as referenced in source).

### Results — SE Loss Estimates and Comparison to Capital Injections
- SE losses (as percent of each bank’s total assets) if Lehman defaulted:
  - Citibank (C): 6.7 percent
  - Wells Fargo (WFC): 5.6 percent
  - Morgan Stanley (MS): 10.0 percent
- TARP capital injections (Date: 11/13/2008; amounts in millions of USD):
  - Citibank (C): Capital Purchase Program 25,000; Targeted Investment Program 20,000; Total Injection 45,000; Total Injection / SE Losses assuming LB default / Total Assets = 2.3; Total Assets = 6.7 (Note: table format preserved in values)
  - Wells Fargo (WFC): Capital Purchase Program 25,000; Targeted Investment Program 0; Total Injection 25,000; Total Injection / SE Losses assuming LB default / Total Assets = 4.0; Total Assets = 5.6
  - Morgan Stanley (MS): Capital Purchase Program 10,000; Targeted Investment Program 0; Total Injection 10,000; Total Injection / SE Losses assuming LB default / Total Assets = 1.0; Total Assets = 10.0
- Interpretation:
  - SE loss estimates were of a similar order of magnitude to capital injections but were larger than actual injections, possibly because market-expected losses were estimated right after Lehman’s default (September 2008), before recapitalization took effect in November 2008.
  - Possible inference: U.S. government recapitalization program was effective in containing default cascades and systemic risk losses, reducing realized needs relative to initial market expectations.

### Decomposition of SE Losses (Probability, Intensity, Contribution)
- Decomposition framework: SE losses decomposed into Probability (Pr), Intensity (In), and Contribution (Co) of conditional losses due to specific defaulting sets.
- Example: Decomposition of SE losses on Citibank conditional on Lehman Brothers defaulting (Table 2)
  - For defaulting sets (notation: defaulting banks listed; non-defaulting entities are underlined and bold):
    - Pr(C|LB): 66.7; 27.5; 1.3; 3.7 (probabilities for the respective defaulting sets)
    - In(C|LB): 0.52; 1.76; 2.02; 3.89 (loss intensity index units)
    - Co(C|LB): 34.5; 48.4; 2.7; 14.5 (contribution in percent of conditional loss)
  - Key insight:
    - Largest contribution to Citibank’s SE losses given LB default is from the conditional loss when LB and MS default but not WFC: Co = 48.4 percent; Pr ≈ 27.5 percent; In = 1.76.
    - Defaulting set LB and WFC but not MS has higher loss intensity (In = 2.02) but very low probability (Pr ≈ 1.3 percent), yielding Co = 2.7 percent.
  - Consistency checks:
    - Co equals Pr multiplied by In for each defaulting set.
    - Sum of Co equals 100 percent.
- Conditional loss scaling:
  - Conditional losses increase as the number of defaulting entities in the defaulting set increases.
  - Table 3 (Conditional Losses under Different Defaulting Sets, in Millions of USD) — values for L(C/D) across different defaulting sets at Dec-08:
    - 41,049.24
    - 191,397.00
    - 303,513.05
    - 460,935.80

### Expected and Book Asset Values — Consistency Checks
- Expected asset values (Table 4a, in Millions USD):
  - Sep-08:
    - C: 2,052,764
    - WFC: 650,171
    - MS: 982,022
  - Dec-08:
    - C: 1,896,820
    - WFC: 1,319,273
    - MS: 653,108
- Book value of assets (Table 4b, in Millions USD; company reports):
  - Sep-08:
    - C: 2,050,131
    - WFC: 622,361
    - MS: 987,403
  - Dec-08:
    - C: 1,938,470
    - WFC: 1,309,639
    - MS: 658,812
- Observation: Expected asset values are consistent with reported book values.

### Implications for the Calibration of Capital Buffers
- Context: Basel III framework requires multiple layers of capital buffers to ensure resilience.
- Baseline requirement: Banks must meet a minimum total capital ratio of 8 percent of risk-weighted assets (RWA) always.
- Additional note: A capital conservation buffer (CCoB) is among the additional capital requirements (further details continue beyond the supplied excerpt).

*Source: wp1849 - Appendix I provides the general formulas when the event S is any combination of k FIs (pdf chapter/section).*

### 2.5 percent of RWA on top of the minimum capital requirement outside periods of

### 2.5 percent of RWA on top of the minimum capital requirement outside periods of stress. The buffer, however, can be drawn down in stress periods.

### Capital buffer types and calibration guidance
- Capital Conservation Buffer (CCoB)
  - Comprised of Common Equity Tier 1 (CET1) and imposes distribution constraints on banks as their capital ratio deteriorates.
  - Banks that draw on this buffer but are not yet in violation of minimum capital requirements can continue their operations but must retain a significant portion of their earnings to rebuild the capital stock.
  - Size referenced: "2.5 percent of RWA on top of the minimum capital requirement outside periods of stress." The buffer can be drawn down in stress periods.
- Countercyclical Capital Buffer (CCyB)
  - Aims to enhance resilience to systemic risks from the financial cycle and reduce procyclicality of bank lending.
  - Can vary between zero and 2.5 percent of RWA.
  - Should build up extra capital in boom times to absorb potential losses in economic downturns.
  - Based on the prevalent state of the macro-financial environment; authorities should increase the CCyB during a lending boom and reduce capital requirements during a contraction.
  - BCBS reference guide uses the aggregate private sector credit-to-GDP gap (Basel Committee on Banking Supervision, 2010).
    - Guide notes: a credit-to-GDP ratio of 10 percentage points or more above trend issues the strongest signal of an impending crisis (in terms of noise-to-signal ratio).
    - Per the BCBS buffer guide formula, when the credit gap breaches a “lower threshold” of 2 percent, a decision to start increasing the buffer could be merited; when it reaches the “upper threshold” of 10 percent, the CCyB should be set at 2.5 percent of RWA. It can also be set higher, based on broader macroprudential considerations (International Monetary Fund, 2014).
- Systemically Important Bank (SIB) capital surcharge
  - Introduced to protect the system from the structural dimension of systemic risk by requiring an additional buffer commensurate to a bank’s contribution to systemic risk.
  - BCBS methodology for G-SIBs (Basel Committee on Banking Supervision, 2013) and for domestically systemically important banks (Basel Committee on Banking Supervision, 2012).
  - Identification uses indicators capturing: size, interconnectedness, level of substitutability, and complexity; G-SIBs add global scope of activities.
  - Banks are ranked and placed in five buckets with a gradual scale of surcharge ranging from 1 to 3.5 percent.

### Use of stress tests and proposed bank-specific approaches
- Macroprudential stress tests represent a useful tool to calibrate CCyB and SIB surcharges.
- Country examples:
  - United Kingdom
    - Authorities intend to set capital requirements for the system-wide CCyB and CCoB, and for the bank-specific Prudential Regulatory Authority (PRA) buffer based in part on stress test results (Bank of England 2015).
    - Sizes: CCoB and CCyB set by the Financial Policy Committee; PRA buffer set by the PRA (both within the BoE).
  - United States
    - Proposal: introduce a bank-specific stress capital buffer (SCB) to replace the 2.5 percent CCoB of the Basel III framework.
    - The SCB would be set at least as high as the CCoB and would be equivalent to the maximum decline of a bank’s tier 1 capital ratio under a severe adverse scenario (Tarullo, 2016).

### Role of SE loss estimation in buffer calibration
- Difficulty: estimating SE losses is challenging due to diverse amplification mechanisms, changing structures, and data limitations.
- Proposed framework and applications:
  - Hiebert and others (2018) augment the framework to quantify the magnitude of SE amplification as a function of the severity of financial imbalances.
  - Authors map levels of financial imbalances to different stages of financial cycles to estimate SE losses conditional on large financial imbalances observed at cycle peaks.
  - This quantification can be useful to ensure CCyB are set to allow banks to withstand such losses (see Figure 6 in source).

### Framework proposition and advantages (macroprudential stress testing for SE)
- Purpose: integrate diverse data and approaches to maximize information content and minimize model error when quantifying losses from SE.
- Key benefits of the proposed encompassing/reduced-form stochastic framework:
  - Combines existing microprudential stress tests with the reduced-form approach to estimate SE losses, leveraging models and expertise already existing in many countries.
  - SE loss estimation can be performed with publicly available data, embedding market perceptions of financial systems’ interconnectedness, including indirect interconnectedness from common risk factors and market price channels.
  - The reduced-form approach does not require explicit modeling of agents’ behavioral reactions, avoiding complex calibration of behaviors.
  - Stochastic structure permits estimation of firms’ asset values conditional on different states of nature and valuations of other firms, allowing quantification of event probabilities and intensity of losses.
  - Multivariate dimension facilitates integration of nonbank financial intermediaries and interactions between banks and nonbanks when quantifying systemic risk amplification losses.
  - Cost-efficient and robust: model is simple and relatively light on data requirements.

### Extensions and further work
- Two extensions highlighted for policymakers:
  - Hiebert and others (2018): augment framework to estimate amplification magnitude conditional on severity of financial imbalances, making SE losses larger (smaller) if imbalances are larger (smaller); useful for CCyB calibration.
  - Espinoza and others (2018): combine a general equilibrium model with the reduced-form approach to develop a systemic risk framework that incorporates systemic risk endogeneity and amplification mechanisms through macroeconomic and systemic risk interactions.
    - Use measurements of SE losses to calibrate parameters of the theoretical model incorporating interbank lending, common asset exposures, and a “Minsky effect.”
    - Calibrations help incorporate non-linear effects (e.g., decrease in prices, increase in probabilities of distress) and changes in behavioral assumptions in times of distress.
    - The combined framework is easily implementable with publicly available data in numerous jurisdictions.

### Appendix I — Conditional expected valuation formulas (generalization)
- General setup:
  - Let the CIMDO posterior density of assets of banks {A1,...,AN} be p(x1,...,xN) over equity annualized returns.
  - Goal: calculate expected value of assets of bank Ai given defaults of banks {Ak1,...,Akl} and other banks not defaulting, i.e., conditioning on event Dj i(k1,...,kl).
- Conditional expected value formula (notation preserved as in source):
  - E0(VAi,t | Dj i(k1,...,kl)) = 1/P(Dj i(k1,...,kl)) E(VAi,t 1Dj i(k1,...,kl))
  - = 1/P(Dj i(k1,...,kl)) ∫∫ VAi,t(x1,...,xN) p(x1,...,xN) ∏ I(xs s∈{j,k1,...,kl} > Xs d) ∏ I(xs < Xs d) d x1 ... d xN
- Decomposition of VAi,t(x1,...,xN):
  - VAi,t(x1,...,xN) = Eqt(x1,...,xN) + Dt(x1,...,xN)
  - Total conditional value expressed as sum of conditional integrals of Eqt and Dt divided by P(Dj i(...)) plus probability-weighted discounted debt terms.
- Debt valuation conditional on default zone:
  - In zone where Ai defaults (xi > Xi d): Dt(x1,...,xN) = (RR) DT e−r(T−t)
  - In zone where Ai does not default (xi < Xi d): Dt(x1,...,xN) = e−r(T−t)
  - RR denotes recovery rate (one minus loss given default).
- Resulting expression (notation preserved):
  - E0(VAi,t | Dj i(k1,...,kl)) = 1/P(Dj i(...)) E q0 ∫∫ e−x1 p(x1,...,xN) ∏ I(...) ∏ I(...) d x1 ... d xN
    + P(Ai ∩ Dj i(...))/P(Dj i(...)) (RR) DT e−r(T−t)
    + P(Āi ∩ Dj i(...))/P(Dj i(...)) DT e−r(T−t)

*Source: IMF working paper content (excerpted).*

### APPENDIX II. GENERALIZATION OF THE SE

### APPENDIX II. GENERALIZATION OF THE SE

### Definition of the vulnerability index assuming any conditioning event
- Assuming the realization of a given event S, the vulnerability index that represents the impact of the assumed default on the bank 퐴ᵢ is defined as:
  - 푉(퐴ᵢ|푆) = 퐿표푠푠_{TS}(퐴ᵢ|푆) / 푇퐴(퐴ᵢ)
- Interpretation:
  - The vulnerability index of bank 퐴ᵢ assuming the default of bank 퐴ⱼ is the total loss under a systemic event 퐿표푠푠_{TS}(퐴ᵢ|푆) of bank 퐴ᵢ divided by its total assets 푇퐴(퐴ᵢ).
  - The higher 푉(퐴ᵢ|푆), the more affected bank 퐴ᵢ is when bank 퐴ⱼ defaults.
  - Computing 푉(·|푆) for every financial institution yields a vulnerability ranking of all FIs in the network conditional on the assumed default S.

### Decomposition of the SE loss
- Setup:
  - Consider a financial network composed of N banks {퐴₁, ..., 퐴_N}.
  - In each subset of the partition of S, each bank (except 퐴ᵢ) is either defaulting or surviving.
  - Define events:
    - 퐷_{Sᵢ}(푘₁,...,푘_l) = S ⋂ ⋂_{푘ᵢ ∈ {푘₁,...,푘_l}} 퐴_{푘ᵢ} ⋂ ⋂_{푘ᵢ ∈ {⟦1,N⟧ \ {푖,S,푘₁,...,푘_l}}} 퐴_{푘ᵢ}
  - Exclude empty subsets from the partition. The collection {퐷_{Sᵢ}(푘₁,...,푘_l)} over all {푘₁,...,푘_l} ∈ 풫(⟦1,N⟧ \ {푖,S}) forms a partition of S; events are disjoint and their union equals S.
- Law of total expectation applied to the vulnerability index:
  - 퐸(푉_{퐴ᵢ}|푆) = ∑_{ {푘₁,...,푘_l} ∈ 풫(⟦1,N⟧ \ {푖, S}) } 푃(퐷_{Sᵢ}(푘₁,...,푘_l)|푆) · 퐸(푉_{퐴ᵢ}|퐷_{Sᵢ}(푘₁,...,푘_l))
  - Here 푃(퐷_{Sᵢ}(푘₁,...,푘_l)|푆) is the probability of subset 퐷_{Sᵢ}(푘₁,...,푘_l) conditional on S.
  - Thus the conditional expected value of bank 퐴ᵢ under S is the probability-weighted average of its values across the partition subsets of S.
- Decomposition for SE losses:
  - Using that {퐷_{Sᵢ}(·)} partitions S and the law of total probability:
    - ∑_{ {푘₁,...,푘_l} ∈ 풫(⟦1,N⟧ \ {푖}) } 푃(퐷_{Sᵢ}(푘₁,...,푘_l)|푆) = 1
  - The SE loss of bank 퐴ᵢ conditional on S is:
    - 퐿표푠푠_{SE}(퐴ᵢ|푆) = 퐸(푉_{퐴ᵢ}) − 퐸(푉_{퐴ᵢ}|푆)
    - = 퐸(푉_{퐴ᵢ}) − ∑_{ {푘₁,...,푘_l} ∈ 풫(⟦1,N⟧ \ {푖, S}) } 푃(퐷_{Sᵢ}(푘₁,...,푘_l)|푆) · 퐸(푉_{퐴ᵢ}|퐷_{Sᵢ}(푘₁,...,푘_l))
    - = ∑_{ {푘₁,...,푘_l} ∈ 풫(⟦1,N⟧ \ {푖, S}) } 푃(퐷_{푗ᵢ}(푘₁,...,푘_l)|푆) · [ 퐸(푉_{퐴ᵢ}) − 퐸(푉_{퐴ᵢ}|퐷_{Sᵢ}(푘₁,...,푘_l)) ]
    - = ∑_{ {푘₁,...,푘_l} ∈ 풫(⟦1,N⟧ \ {푖, S}) } 푃(퐷_{푗ᵢ}(푘₁,...,푘_l)|푆) · 퐿표푠푠_{SE}(퐴ᵢ|퐷_{Sᵢ}(푘₁,...,푘_l))

### Indexes of the decomposition of the SE loss
- Aggregate contribution index of a defaulting set (that includes the initially defaulting bank) to the SE loss of bank 퐴ᵢ conditional on S:
  - 퐶표(퐷_{Sᵢ}(푘₁,...,푘_l)|푆) = [ 푃(퐷_{Sᵢ}(푘₁,...,푘_l)|푆) · 퐿표푠푠_{SE}(퐴ᵢ|퐷_{Sᵢ}(푘₁,...,푘_l)) ] / 퐿표푠푠_{SE}(퐴ᵢ|푆)
- Interpretation and policymaker relevance:
  - 퐶표(·|푆) represents the relative contribution of a particular defaulting set to the conditional SE loss of the tested bank 퐴ᵢ, given S.
  - This measure helps identify connecting entities that induce a large share of SE losses to the tested bank under the given event S.
- Two drivers of large contributions (as highlighted in the four banks example):
  - Likelihood driver:
    - The considered defaulting set may be likely to materialize given S. This is assessed by the conditional probability:
      - 푃(퐷_{Sᵢ}(푘₁,...,푘_l)|푆)
  - Intensity driver:
    - The considered defaulting set may inflict large losses to the tested bank conditional on S. This is assessed by the intensity ratio:
      - 퐼푛(퐷_{Sᵢ}(푘₁,...,푘_l)|푆) = 퐿표푠푠_{SE}(퐴ᵢ|퐷_{Sᵢ}(푘₁,...,푘_l)) / 퐿표푠푠_{SE}(퐴ᵢ|푆)
    - Properties of the intensity ratio:
      - The ratio lies between zero and infinity.
      - The larger the ratio, the more intense the impact of that defaulting set on the tested bank (conditional on S).

*Source: APPENDIX II. GENERALIZATION OF THE SE, wp1849*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp1849.pdf_