## 1. Comparison of median estimates from the symmetric and asymmetric CoVaR models.........27

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---

### Major themes
- Comparison of median estimates from the symmetric and asymmetric CoVaR models.
- Cross-sectional median estimates of the decile-based coefficients.
- Presentation of sample descriptives and descriptive statistics for economic and financial state variables.
- Reporting of median and decile-sorted estimates for the symmetric and asymmetric CoVaR across different firm groupings, including size, liabilities, and bank types.

### Modeling Systemic Risk: CoVaR
- Definitions and notation:
  - X_{t,S} and X_{t,i} denote simple returns of the whole financial system and of the individual institution, respectively.
  - For probability τ∈(0,1), VaR_{P,τ,t} is the τ·100 percent VaR; CoVaR_{S|i,t}(τ,τ′) is the τ-quantile of X_{t,S} conditional on X_{t,i}=VaR_{i,τ′,t}.
  - Under τ′=τ, ΔCoVaR_{S|i,t}(τ) = CoVaR_{S|i,t}(τ,τ) − CoVaR_{S|i,t}(τ,0.5).
- Adrian and Brunnermeier (2011) specification:
  - Conditional quantile linear form: Q_{X_{t,S}}(τ|·) = Z_{t−1}'θ_M(τ) + θ_i(τ) X_{t,i}.
  - CoVaR prediction: \hat{CoVaR}_{S|i,t}(τ) = Z_{t−1}'\hat{θ}_M(τ) + \hat{θ}_i(τ) \widehat{VaR}_{i,τ,t}.
- Limitation:
  - Linear dependence assumes symmetric feed-through of positive and negative shocks and may understate downside comovement.

### Asymmetric CoVaR (proposed extension)
- Construction:
  - Partition individual returns into negative and non-negative signed processes: X^-_{t,i} = X_{t,i} I(X_{t,i}<0), X^+_{t,i} = X_{t,i} I(X_{t,i}≥0).
  - Conditional mean example: X_{t,S} = β + M'_{t−1}γ + δ_{1,i} X^-_{t,i} + δ_{2,i} X^+_{t,i} + ε_t.
  - Sign-dependent volatility allowed: σ_t(X_{t,i}) = σ_{0,i} + σ_{1,i} X^-_{t,i} + σ_{2,i} X^+_{t,i}.
- Quantile specification and estimation:
  - Asymmetric CoVaR quantile form is linear in parameters: Q_{X_{t,S}}(τ|·) = 1_t' θ(τ) with θ(τ) including δ^-(τ) and δ^+(τ).
  - Parameters estimated by linear quantile regression (Koenker and Bassett (1978)).
  - Wald test for symmetry: H0: δ^-(τ) = δ^+(τ), statistic asymptotically χ^2(1).
  - Covariance matrix estimated via sandwich-type estimator combining kernel density and heteroscedasticity-consistent covariance.

### Data
- Quarterly balance-sheet data source: Federal Reserve Bank of Chicago Bank Regulatory Database.
- Sample period: Q1 1990 through Q4 2010.
- Raw sample:
  - 32,204 panel-data observations grouped into 791 BHC and 65 CB.
- Filtered sample (minimum 500 weeks traded requirement):
  - 21,786 panel-data observations from 340 BHC and 25 CB.
- Dependent variable construction:
  - Weekly returns of market-valued total assets for banks; smoothing of quarterly leverage via cubic spline interpolation (robust to alternative constant-leverage approach).
- System portfolio constructions:
  - Single-index system: value-weighted average of all banks in total sample (weights ω_{t,j}=A_{t−1,j}/Σ_s A_{t−1,s}).
  - Multiple-index system: for each bank i, system excludes bank i (weights ω'_{t,j} set to zero if j=i).
- Predictors (Z_t): VIX, Market Return (S&P500), Δ 3-month T-bill (ΔT-bill), Yield Slope (10yr − 3mo), Default Premium (10yr Baa − 10yr Treasury), NBER recessions dummy, Financial Recession dummy (Aug 2007–Mar 2009).

### Downside comovement — Main empirical results
- Estimation setup:
  - VaR and CoVaR estimated weekly at τ ∈ {0.01, 0.05} via quantile regressions.
- Representative median coefficient estimates across banks (multiple-index system):
  - Symmetric CoVaR: median θ_i(τ) = 0.086 at τ=0.01; 0.094 at τ=0.05.
  - Asymmetric CoVaR: median δ^-(τ) = 0.372 at τ=0.01; 0.337 at τ=0.05.
  - Asymmetric CoVaR: median δ^+(τ) = 0.048 at τ=0.01; 0.030 at τ=0.05.
- Predictive variables:
  - VIX is the most effective predictor of tail conditional distribution; influence increases as τ decreases.
- Test rejection frequencies (sample frequencies of rejecting no-interdependence at 95%):
  - Symmetric model H0:Ind rejected in 42.47% of banks at τ=0.01 and 57.53% at τ=0.05.
  - Asymmetric model H0:Ind rejected in 88.49% at τ=0.01 and 90.68% at τ=0.05.
  - Asymmetric H0:Loss rejection: 82.47% (τ=0.01), 88.49% (τ=0.05).
  - Asymmetric H0:Sym rejection: 66.19% (τ=0.01), 73.42% (τ=0.05).
- Symmetry:
  - Symmetry restriction H0: δ^-(τ)=δ^+(τ) is widely rejected (unconditional rejection ratio ≈ 70%).
- Size dependence (multiple-index approach):
  - Symmetric θ_i(τ) increases with size: top decile θ ≈ 0.150 at τ=0.01 and 0.276 at τ=0.05; bottom decile θ ≈ 0.060 at τ=0.01 and 0.041 at τ=0.05.
  - Asymmetric loss-related δ^- much larger at top decile: δ^- ≈ 0.816 at τ=0.01 versus δ^- ≈ 0.135 at bottom decile (τ=0.01).
  - Rejection rates of H0:Ind, H0:Loss, and H0:Sym increase with bank size.
- Magnitude comparisons and examples:
  - Average δ^-(τ) is roughly 4 times larger than symmetric θ_i(τ) (e.g., 0.372 vs 0.086 at τ=0.01).
  - Citigroup median weekly ΔCoVaR at τ=0.01: asymmetric median = −0.0565; symmetric = −0.008 (factor ≈ 7 difference).
  - Relative impact and too-big-to-fail:
    - For a median-sized bank, relative impact ratio is sevenfold (negative vs average).
    - Tail interdependence ratio: 2.2 for bottom-size decile vs 5.4 for top-size decile.

### Discussion — implications of misspecification
- Symmetric CoVaR limitations:
  - Pools positive and negative effects, yielding downward-biased estimates of tail comovement and conservative inference about interconnectedness.
  - Bias is systematic and correlated with bank size — larger banks' systemic contributions more severely underestimated.
- Asymmetric CoVaR advantages:
  - Captures sign-dependent comovements in conditional mean and volatility.
  - Produces larger, statistically significant loss-related coefficients (δ^-) and higher rejection rates for nulls of no-interdependence.
  - Aligns with empirical findings that correlations increase during negative market movements.
- Economic and regulatory mechanisms generating asymmetry:
  - Fire-sales, funding contraction, investor confidence spillovers, non-linear risk-weight transitions under standardized regulatory approaches, and provisioning behavior amplify downside comovements.

### Robustness checks
- Institution type segmentation:
  - Bank Holding Companies (BHC) vs Commercial Banks (CB): same qualitative pattern; CBs show smaller systemic importance and lower rejection ratios.
- Nonlinear extensions:
  - Piecewise linear models with multiple thresholds (decile-based specification) indicate most left-tail comovement is driven by negative-return deciles and that asymmetric CoVaR captures first-order nonlinearity effectively.
- Alternative portfolio and smoothing choices:
  - Using equity returns instead of balance-sheet derived returns yields the same asymmetric model advantage.
  - Weighting system returns by liabilities instead of assets produces similar outcomes.
  - Cubic spline smoothing vs linear interpolation for quarterly-to-weekly leverage: results robust.

### Key quantitative findings and statistics
- Sample and data:
  - Total sample: 32,204 observations; 791 BHC; 65 CB.
  - Filtered sample (≥500 weeks): 21,786 observations; 340 BHC; 25 CB.
  - Sample period: Q1 1990 through Q4 2010.
- Median CoVaR parameters (multiple-index):
  - Symmetric θ_i(τ): 0.086 (τ=0.01), 0.094 (τ=0.05).
  - Asymmetric δ^-(τ): 0.372 (τ=0.01), 0.337 (τ=0.05).
  - Asymmetric δ^+(τ): 0.048 (τ=0.01), 0.030 (τ=0.05).
- Test rejection frequencies:
  - Symmetric H0:Ind rejection: 42.47% (τ=0.01), 57.53% (τ=0.05).
  - Asymmetric H0:Ind rejection: 88.49% (τ=0.01), 90.68% (τ=0.05).
  - Asymmetric H0:Loss rejection: 82.47% (τ=0.01), 88.49% (τ=0.05).
  - Asymmetric H0:Sym rejection: 66.19% (τ=0.01), 73.42% (τ=0.05).
- Size-related medians (example, multiple-index):
  - Symmetric top-decile θ: 0.150 (τ=0.01), 0.276 (τ=0.05).
  - Asymmetric top-decile δ^-: 0.816 (τ=0.01).
- Representative ΔCoVaR example:
  - Citigroup median weekly ΔCoVaR at τ=0.01: asymmetric = −0.0565; symmetric = −0.008.

### Policy recommendations and implications
- Risk measurement:
  - Account explicitly for asymmetric tail comovements (negative vs positive returns) when estimating systemic risk contributions (CoVaR).
- Prudential policy:
  - Recognize systematic underestimation of systemic contributions by linear/symmetric models, particularly for large banks; support regulatory measures that impose higher loss-absorbency on systemically important institutions.
- Resolution and supervision:
  - Complement improved risk measurement with structural policy tools: higher capital buffers for systemic institutions, resolution planning for large institutions, and enhanced supervisory oversight.
- Surveillance and stress scenarios:
  - Use asymmetric CoVaR in systemic risk monitoring and stress-testing frameworks to capture nonlinear spillovers during deleveraging and elevated balance-sheet stress.

### Conclusion
- The asymmetric CoVaR is a tractable extension of Adrian and Brunnermeier (2011) that captures sign-dependent spillovers in both conditional mean and volatility, producing materially larger and more statistically significant measures of downside systemic contributions.
- Misspecifying CoVaR as symmetric leads to conservative underestimation of systemic risk, with distortions increasing with institution size — a critical concern for macroprudential policy directed at mitigating too-big-to-fail externalities.
- The asymmetric CoVaR provides practical gains in systemic risk quantification and supports policy measures (higher loss absorbency and robust resolution planning) aimed at strengthening financial stability.

*Source: IMF Working Paper content (References and body text as provided in the source PDF).*

### 1. Comparison of median estimates from the symmetric and asymmetric CoVaR models.........27

### 1. Comparison of median estimates from the symmetric and asymmetric CoVaR models.........27

### Major themes
- Comparison of median estimates from the symmetric and asymmetric CoVaR models.
- Cross-sectional median estimates of the decile-based coefficients.
- Presentation of sample descriptives and descriptive statistics for economic and financial state variables.
- Reporting of median and decile-sorted estimates for the symmetric and asymmetric CoVaR across different firm groupings, including size, liabilities, and bank types.

### Included tables and listings
- 1. Sample descriptives for the total and the filtered samples..........................................29
- 2. Descriptive statistics for economic and financial state variables..................................30
- 3. Median estimates for the symmetric and asymmetric CoVaR.....................................31
- 4. Estimates across size-sorted deciles for the symmetric and asymmetric CoVaR...............32
- 5. Estimates across liabilities-sorted deciles for the symmetric and asymmetric CoVaR.........33
- 6. Estimates across BHCs and CBs for the symmetric and asymmetric CoVaR...................34

_Unit: 1. Comparison of median estimates from the symmetric and asymmetric CoVaR models.........27_

### References...................................................................................................35

### _wp12152 - References...................................................................................................35

### Introduction
- Context: Post-2008 collapse of Lehman Brothers prompted massive government bailouts and a revamp of global capital and liquidity rules to limit systemic contagion and moral hazard.
- Objective: Assess the CoVaR framework of Adrian and Brunnermeier (2011) and propose an extension that captures nonlinear tail comovements between individual bank returns and financial system returns (termed "asymmetric CoVaR").

### Modeling Systemic Risk: CoVaR
- Key definitions and notation:
  - X_{t,S} and X_{t,i} denote simple returns of the whole financial system and of the individual institution, respectively.
  - For probability τ∈(0,1), VaR_{P,τ,t} is the τ·100 percent VaR; CoVaR_{S|i,t}(τ,τ′) is the τ-quantile of X_{t,S} conditional on X_{t,i}=VaR_{i,τ′,t}.
  - Under τ′=τ, ΔCoVaR_{S|i,t}(τ) = CoVaR_{S|i,t}(τ,τ) − CoVaR_{S|i,t}(τ,0.5).
- Adrian and Brunnermeier (2011) specification:
  - Conditional quantile linear form: Q_{X_{t,S}}(τ|·) = Z_{t−1}'θ_M(τ) + θ_i(τ) X_{t,i}.
  - CoVaR prediction: \hat{CoVaR}_{S|i,t}(τ) = Z_{t−1}'\hat{θ}_M(τ) + \hat{θ}_i(τ) \widehat{VaR}_{i,τ,t}.
- Limitation: Linear dependence assumes symmetric feed-through of positive and negative shocks and may understate downside comovement.

### Asymmetric CoVaR
- Proposed generalization (preserves tractability):
  - Partition individual returns into negative and non-negative signed processes: X^-_{t,i} = X_{t,i} I(X_{t,i}<0), X^+_{t,i} = X_{t,i} I(X_{t,i}≥0).
  - Conditional mean specification (example): X_{t,S} = β + M'_{t−1}γ + δ_{1,i} X^-_{t,i} + δ_{2,i} X^+_{t,i} + ε_t, with volatility allowed to depend on sign via σ_t(X_{t,i}) = σ_{0,i} + σ_{1,i} X^-_{t,i} + σ_{2,i} X^+_{t,i}.
  - Asymmetric CoVaR quantile form (linear in parameters for estimation):
    - Q_{X_{t,S}}(τ|·) = 1_t' θ(τ) where θ(τ) collects parameters including δ^- (τ) and δ^+ (τ).
- Estimation and inference:
  - Parameters estimated by linear quantile regression (Koenker and Bassett (1978)).
  - Wald test for symmetry: H0: δ^-(τ) = δ^+(τ) with statistic asymptotically χ^2(1).
  - Covariance matrix estimated via sandwich-type estimator combining kernel density and heteroscedasticity-consistent covariance.

### Data
- Quarterly balance-sheet data source: Federal Reserve Bank of Chicago Bank Regulatory Database.
- Sample period: Q1 1990 through Q4 2010.
- Raw sample:
  - 32,204 panel-data observations grouped into 791 BHC and 65 CB.
- Filtered sample (minimum 500 weeks traded requirement):
  - 21,786 panel-data observations from 340 BHC and 25 CB.
- Dependent variable construction:
  - Weekly returns of market-valued total assets for banks; smoothing of quarterly leverage via cubic spline interpolation (robust to alternative constant-leverage approach).
- System portfolio constructions:
  - Single-index system: value-weighted average of all banks in total sample (weights ω_{t,j}=A_{t−1,j}/Σ_s A_{t−1,s}).
  - Multiple-index system: for each bank i, system excludes bank i (weights ω'_{t,j} set to zero if j=i).
- Predictors (Z_t): VIX, Market Return (S&P500), Δ 3-month T-bill (ΔT-bill), Yield Slope (10yr − 3mo), Default Premium (10yr Baa − 10yr Treasury), NBER recessions dummy, Financial Recession dummy (Aug 2007–Mar 2009).

### Downside Comovement in the U.S. Banking Industry — Main empirical results
- VaR and CoVaR estimation done weekly at τ ∈ {0.01, 0.05} via quantile regressions.
- Representative median coefficient estimates across banks (multiple-index system):
  - Symmetric CoVaR: median θ_i(τ) = 0.086 at τ=0.01; 0.094 at τ=0.05.
  - Asymmetric CoVaR: median δ^-(τ) = 0.372 at τ=0.01; 0.337 at τ=0.05. Median δ^+(τ) = 0.048 at τ=0.01; 0.030 at τ=0.05.
- Predictive variables:
  - VIX is the most effective predictor of tail conditional distribution; influence increases as τ decreases.
- Sample frequencies of rejecting no-interdependence (H0: CoVaR parameter(s)=0) at 95%:
  - Symmetric model H0:Ind rejected in 42.47% of banks at τ=0.01 and 57.53% at τ=0.05.
  - Asymmetric model H0:Ind rejected in 88.49% at τ=0.01 and 90.68% at τ=0.05.
- Symmetry restriction H0: δ^-(τ)=δ^+(τ) is widely rejected (unconditional rejection ratio ≈ 70%).
- Size dependence (total assets sorted into deciles) — multiple-index approach highlights:
  - Symmetric CoVaR θ_i(τ) increases with size: top decile θ ≈ 0.150 at τ=0.01 and 0.276 at τ=0.05; bottom decile θ ≈ 0.060 at τ=0.01 and 0.041 at τ=0.05.
  - Asymmetric CoVaR loss-related δ^-(τ) is much larger at top decile: δ^- ≈ 0.816 at τ=0.01 versus δ^- ≈ 0.135 at bottom decile (τ=0.01).
  - Rejection rates of H0:Ind, H0:Loss, and H0:Sym increase with bank size.
- Bias and magnitudes:
  - Average δ^-(τ) is roughly 4 times larger than symmetric θ_i(τ) (e.g., 0.372 vs 0.086 at τ=0.01).
  - Under symmetric model, median weekly ΔCoVaR for Citigroup at τ=0.01: asymmetric median = −0.0565; symmetric model yields −0.008 (factor ≈ 7 difference).
  - Relative impact ratios and too-big-to-fail:
    - For a median-sized bank, relative impact ratio is sevenfold (negative vs average).
    - Tail interdependence ratio: 2.2 for bottom-size decile vs 5.4 for top-size decile (conveying larger downside amplification for large banks).

### Discussion — implications of misspecification
- The symmetric (linear) CoVaR:
  - Pools positive and negative effects, yielding downward-biased estimates of tail comovement and conservative inference about interconnectedness.
  - Bias is systematic and correlated with bank size — larger banks' systemic contributions are more severely underestimated.
- The asymmetric CoVaR:
  - Captures sign-dependent comovements in both conditional mean and volatility.
  - Produces larger, statistically significant loss-related coefficients (δ^-) and higher rejection rates for nulls of no-interdependence.
  - Outperforms symmetric model and aligns with empirical findings that correlations increase during negative market movements.
- Economy and regulatory mechanisms that generate asymmetry:
  - Fire-sales, funding contraction, investor confidence spillovers, non-linear risk-weight transitions under standardized regulatory approaches, and provisioning behavior amplify downside comovements.

### Robustness checks
- Segmenting sample by institution type:
  - Bank Holding Companies (BHC) vs Commercial Banks (CB): same qualitative pattern; CBs show smaller systemic importance and lower rejection ratios, consistent with smaller size and business model differences.
- Nonlinear extensions:
  - Piecewise linear models with multiple thresholds (decile-based specification) estimated; results indicate most left-tail comovement is driven by negative-return deciles and that asymmetric CoVaR captures first-order nonlinearity effectively.
- Alternative portfolio and smoothing choices:
  - Using equity returns (stock returns) instead of balance-sheet derived returns yields the same asymmetric model advantage.
  - Weighting system returns by liabilities instead of assets produces similar outcomes.
  - Cubic spline smoothing vs linear interpolation for quarterly-to-weekly leverage: results robust.

### Key quantitative findings and statistics
- Sample and data:
  - Total sample: 32,204 observations; 791 BHC; 65 CB.
  - Filtered sample (≥500 weeks): 21,786 observations; 340 BHC; 25 CB.
  - Sample period: Q1 1990 through Q4 2010.
- Median CoVaR parameters (multiple-index):
  - Symmetric θ_i(τ): 0.086 (τ=0.01), 0.094 (τ=0.05).
  - Asymmetric δ^-(τ): 0.372 (τ=0.01), 0.337 (τ=0.05).
  - Asymmetric δ^+(τ): 0.048 (τ=0.01), 0.030 (τ=0.05).
- Test rejection frequencies (median/sample):
  - Symmetric H0:Ind rejection: 42.47% (τ=0.01), 57.53% (τ=0.05).
  - Asymmetric H0:Ind rejection: 88.49% (τ=0.01), 90.68% (τ=0.05).
  - Asymmetric H0:Loss rejection: 82.47% (τ=0.01), 88.49% (τ=0.05).
  - Asymmetric H0:Sym rejection: 66.19% (τ=0.01), 73.42% (τ=0.05).
- Size-related medians (example, multiple-index):
  - Symmetric top-decile θ: 0.150 (τ=0.01), 0.276 (τ=0.05).
  - Asymmetric top-decile δ^-: 0.816 (τ=0.01).
- Representative ΔCoVaR example:
  - Citigroup median weekly ΔCoVaR at τ=0.01: asymmetric = −0.0565; symmetric = −0.008.

### Policy recommendations and implications
- Risk measurement:
  - Account explicitly for asymmetric tail comovements (negative vs positive returns) when estimating systemic risk contributions (CoVaR).
- Prudential policy:
  - Recognize systematic underestimation of systemic contributions by linear/symmetric models, particularly for large banks; this supports regulatory measures that impose higher loss-absorbency on systemically important institutions.
- Resolution and supervision:
  - Complement improved risk measurement with structural policy tools already recommended or implemented: higher capital buffers for systemic institutions, resolution planning (e.g., Dodd-Frank resolution plans for institutions with $50 billion+ in total assets), and enhanced supervisory oversight.
- Surveillance and stress scenarios:
  - Use asymmetric CoVaR in systemic risk monitoring and stress-testing frameworks to capture nonlinear spillovers during deleveraging and elevated balance-sheet stress.

### Conclusion
- The asymmetric CoVaR is a tractable extension of Adrian and Brunnermeier (2011) that captures sign-dependent spillovers in both conditional mean and volatility, producing materially larger and more statistically significant measures of downside systemic contributions.
- Misspecifying CoVaR as symmetric leads to conservative underestimation of systemic risk, with distortions increasing with institution size — a critical concern for macroprudential policy directed at mitigating too-big-to-fail externalities.
- The asymmetric CoVaR provides practical gains in systemic risk quantification and supports policy measures (higher loss absorbency and robust resolution planning) aimed at strengthening financial stability.

*Source: IMF Working Paper content (References and body text as provided in the source PDF).*

### References

### _wp12152 - References

### Systemic risk, CoVaR, and systemic-importance assessment
- Acharya, V.V., Pedersen, L.H., Philippon, T. and Richardson, M. (2010). ìMeasuring systemic riskî. Federal Reserve Bank of Cleveland, Working Paper 10-02.
- Adrian, T. and Brunnermeier, M. (2011): ìCoVaRî, Princeton University, Working Paper.
- Brownlees, C. T. and Engle, R.F. (2011). ìVolatility, correlation and tails for systemic risk measurementî. New York University, Working Paper.
- Huang, X., Zhou, H. and Zhu, H. (2009). ìA framework for assessing the systemic risk of major Önancial institutionsî. Journal of Banking and Finance,33, 2036-2049.
- Bank for International Settlements, (2011). ìGlobal systemically important banks: Assessment methodology and the additional loss absorbency requirementî.
- Goodhart, C. and Segoviano, M. (2009). ìBanking stability measures,îInternational Monetary Fund Working Paper 09/4.
- Zhou, C. (2010). ìAre banks too big to fail? Measuring systemic importance of Önancial institutionsî. International Journal of Central Banking,6 (4), 205-250.
- LÛpez-Espinosa, G., Moreno, A., Rubia, A., and Valderrama, L. (2012). ìShort-term wholesale funding and systemic risk: A global CoVaR approachî. Journal of Banking and Finance, 10.1016/j.jbankÖn.2012.04.020.

### Market liquidity, funding liquidity, and connectedness
- Brunnermeier, M., and Pedersen, L. (2009), ìMarket liquidity and funding liquidityî, Review of Financial Studies22(6), 2201-2238.
- Diebold, F.X., and Yilmaz, K. (2011). ìOn the network topology of variance decompositions: Measuring the connectedness of Önancial Örmsî, University of Pennsylvania Working Paper.
- Andersen, T. G., Bollerslev, T., Diebold, F. X. and Vega, C. (2007) ìReal-time price discovery in stock, bond and foreign exchange marketsî, Journal of International Economics, 73, 251-277.
- Beber, A., and Brandt M.W. (2010). ìWhen it cannot get better or worse: The asymmetric impact of good and bad news on bond returns in expansions and recessionsî, Review of Finance,14, 119-155.

### Volatility, tails, downside risk, and conditional heteroskedasticity
- Black, F. (1976). ìStudies of stock market volatility changesî.1976 Proceedings of the American Statistical Association, Business and Economic Statistics Section, 177-181.
- Bekaert, G., and Wu, G. (2000), ìAsymmetric volatility and risk in equity marketsî. Review of Financial Studies, 13(1), 1-42.
- Christie, A.A. (1982). ìThe stochastic behavior of common stock variances: Value, leverage and interest rate e§ectsî. Journal of Financial Economics, 10, 407-432.
- French, K.R., Schwert, G.W., and Stambaugh, R. (1987). ìExpected stock return and volatilityî. Journal of Financial Economics, 19, 3-29.
- McQueen, G., and Vorkink, K. (2004). ìWhence GARCH? A preference-based explanation for conditional volatilityî. Review of Financial Studies, 17(4), 914-949.
- Nelson, D.B. (1991). ìConditional heteroskedasticity in asset returns: A new approachî. Econometrica, 59(2), 347-390.
- Brownlees, C. T. and Engle, R.F. (2011). ìVolatility, correlation and tails for systemic risk measurementî. New York University, Working Paper.

### Quantile, extreme-value, and tail-risk methods
- Chernozhukov, V. (2005), ìExtremal quantile regressionî. The Annals of Statistics, 33(2), 806-839.
- Chernozhukov, V. and Umantsev, L. (2001). ìConditional Value-at-Risk: Aspects of modeling and estimation.î Empirical Economics, 26(1), 271-292.
- Koenker, R. (2005). Quantile Regression. Econometric Society Monograph Series, Cambridge University Press: Cambridge.
- Koenker, R. and Bassett, G. (1978). ìRegression Quantiles.î Econometrica, 46(1), 33-50.
- Taylor, J.W. (1999). ìA Quantile regression approach to estimating the distribution of multiperiod returns.î Journal of Derivatives, 7(1), 64-78.
- Longin, F., and Solnik, B. (2001). ìExtreme correlation of international equity marketsî. Journal of Finance, 56, 649-676.
- McNeil, A.J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press: Princeton.

### Asset pricing, downside measures, and behavioral foundations
- Ang, A., Chen, J. and Xing, Y. (2006): ìDownside Riskî. Review of Financial Studies,19, 1191-1239.
- Bali, T.G., Demirtas, K.O., and Levy, H. (2009) ìIs there an intertemporal relation between downside risk and expected returns?î. Journal of Financial and Quantitative Analysis, 44(4), 883ñ909.
- Barberis, N., Huang, M., and Santos, T., (2001). ìProspect theory and asset pricesî. Quarterly Journal of Economics,116, 1ñ53.
- Bawa, V., and Lindenberg, E. (1977). ìCapital market equilibrium in a mean-lower partial moment frameworkî. Journal of Financial Economics, 5, 189ñ200.
- Harlow, V., and Rao, R. (1989). ìAsset pricing in a generalized mean-lower partial moment framework: Theory and evidenceî. Journal of Financial and Quantitative Analysis, 24, 285ñ311.
- Hogan, W., and Warren, J. (1974). ìToward the development of an equilibrium capital-market model based on semivarianceî. Journal of Financial and Quantitative Analysis, 9,1ñ11.
- Gul, F. (1991). ìA theory of disappointment aversionî. Econometrica, 59(3), 667-689.
- Berkelaar, A., and Kouwenberg, R. (2009). ìFrom boom ëtil bust: How loss aversion a§ects asset pricesî. Journal of Banking and Finance 33, 1005-1013.

### Historical crisis propagation and macroprudential policy
- Bernanke, B.S. (1983). ìNonmonetary e§ects of the Önancial crisis in the propagation of the Great Depressionî. American Economic Reviews73(3), 257ñ76.
- Nier, E. (2011). ìMacroprudential policyñTaxonomy and challengesî, National Institute Economic Review 216, April.
- International Monetary Fund (2012). Global Financial Stability Report, World Economic and Financial Surveys (Washington: International Monetary Fund.

*Reference list from _wp12152 - References*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12152.pdf_