Trading book and credit risk : bending the binds

Transcription

1 Trading book and credit risk : bending the binds Stéphane THOMAS based on a joint work with J-P. Laurent & M. Sestier 9th Financial Risks International Forum - Paris - 22th March / 26

2 Contents Basel recommandations on credit risk 1 Basel recommandations on credit risk Credit risk in Basel 2.5 Credit risk in Basel III FRTB 2 Credit risk models Correlation modelling Hoeffding and risk decomposition 3 EU Corporate exposures: long only and long/short portfolios Impact on 99.9% VaR Drivers of risk (systematic vs idiosyncratic) 2 / 26

3 The RWA conundrum I Credit risk in Basel 2.5 Credit risk in Basel III FRTB Is a simple and standardised indicator a trustworthy one? What about a fine-tuned and institution-specific one? Source: Haldane s speech at the Fed of Atlanta, US (9th April 2013) [1] 3 / 26

4 The RWA conundrum II Credit risk in Basel 2.5 Credit risk in Basel III FRTB Basel 2.5 IRC RWA show the largest variability Source: Haldane s speech at the Fed of Atlanta, US (9th April 2013) [1] 4 / 26

5 Credit risk in Basel 2.5 Credit risk in Basel III FRTB Credit risk in Basel 2.5 (IRC) and RWA variability Basel framework : the Risk Weighted Assets (RWA) Minimum Capital Requirement = X% RWA (1) RWA for credit risk in the trading book: Incremental Risk Charge (IRC) BCBS - Basel 2.5 (2009) [2] No prescribed model (internal, often multi-factorial model for the default correlation). RWA variability tackled - Within the regulation philosophy, variability of RWA among financial institutions should mostly stem from discrepancies in activity, local jurisdictions or risk profiles. - It appears in fact that internal models implementations are in cause, especially for the IRC calculation. BCBS - RCAP Trading Book (2013) [3, 4] 5 / 26

6 Credit risk in Basel 2.5 Credit risk in Basel III FRTB RWA variability : Hypothetical Portfolio Exercices BCBS (2014) [5] analyses that most banks currently use an IRC model with 3 or less factors, and only 3% have more than 3 DULLMAN et al. (2007) [6] empirically show that there surely is a complex interaction between correlations and PD when varying the number of factors Source : Second report on RWA in the trading book. BCBS - Regulatory Consistency Assessment Program (2013) [3] 6 / 26

7 Credit risk in Basel 2.5 Credit risk in Basel III FRTB Basel III FRTB: the Default Risk Charge (DRC) Improving the RWA comparability among financial institutions Prescriptive constraints on the modelling choices for internal models Basel III FRTB, RWA for credit risk: Default Risk Charge (DRC) BCBS - Fundamental Review of the Trading Book (2012, 2013, 2015) [7, 8, 9] PD, LGD, default correlation matrix Based on a prescribed two-factor model for the default correlation Two papers in the literature addressing these questions LAURENT, SESTIER and THOMAS (2015) [10]: focuses on the correlation matrix estimation through a statistical approach WILKENS and PREDESCU (2015) [11]: provides a full calibration methodology through an economic approach 7 / 26

8 Portfolio loss assumptions Credit risk models Correlation modelling Hoeffding and risk decomposition One period portfolio loss L = k EAD k LGD k DefaultIndicator k (2) - Exposures (EAD) and Losses Given Default (LGD) assumed constant for simplicity. In this study, we then focus on correlation modelling to grasp the causes of potential RWA variability Trading book inventories - Exposures may be long (sign +) or short (sign -) - CDS or bond exposures Structural-type default model - Default occurs if a latent variable, X k (creditworthiness), lies below a threshold: DefaultIndicator k = 1 {Xk threshold k } (3) 8 / 26

9 Prescribed two-factor model Credit risk models Correlation modelling Hoeffding and risk decomposition The Committee has decided to develop a more prescriptive DRC charge in the modelsbased framework. Banks using the internal model approach to calculate a default risk charge must use a two-factor default simulation model, which the Committee believes will reduce variation in market risk-weighted assets but be sufficiently risk sensitive as compared to multifactor models. BCBS (2013) [8] Factor models X k = β k Z + 1 β k β kɛ k (4) - Z N(0, Id J ): systematic factor. - ɛ k N(0, 1) : specific risk. - β R K,J : factor loadings. - threshold k = Φ 1 (p k ) with p k the default probability of the obligor k and Φ the Gaussian cdf. MERTON (1974) [12], BCBS (IRB) (2004) [13], ROSEN & SAUNDERS (2010) [14]. not prescriptive: could be latent (endogeneous) or observable (exogeneous) factors 9 / 26

10 Prescribed calibration data Credit risk models Correlation modelling Hoeffding and risk decomposition Default correlations must be based on credit spreads or on listed equity prices. BCBS (2015) [15] correlations [should] be calibrated over a one-year stress period [... ] using [... ] annual co-movements [... ] which took place within the last ten years BCBS - FAQ QIS Market Risk (2015) [16] We chose to rely on daily co-movements taken over a one-year stress-period, because of data availability (mostly for CDS) and cliff effects considerations Remark The BCBS letsthe question of noisy information aside, despite that it is a central source of variability (see Michaud [1989], Laloux et al. [1999], Papp et al. [2005]) Let s consider X R K T the historical sample of centered returns (equity prices or CDS spreads), along three specifications: Sample covariance matrix : Σ Sample = T 1 XX t Shrinked covariance matrix : Σ Shrinkage = ασ FactorModel + (1 α)σ Sample Initial correlation matrix : C 0 = (diag(σ)) 1/2 Σ(diag(Σ)) 1/2 10 / 26

11 Calibration approach Credit risk models Correlation modelling Hoeffding and risk decomposition No guidance by the BCBS on how to pass from a (J > 2)-factor structure to a (J = 2)-factor structure Economic approach: - exogeneous variables only - system-wise - need for an equity return model Statistical approach: - exogeneous or endogeneous variables - portfolio-wise - no need for an equity return model Nearest correlation matrix with a two-factor structure { arg minβ f obj (β) = C(β) C 0 F subject to β Ω = {β R K 2 β k β k 1, k = 1,..., K} Constraint ensures that C(β) = ββ t + diag(id ββ t ) is positive semi-definite. PCA-based method and Spectral projected gradient (SPG) method ANDERSEN et al. (2003) [17], BIRGIN et. al (2000, 2001) [18, 19] 11 / 26

12 Credit risk models Correlation modelling Hoeffding and risk decomposition Unconstrained correlation matrix and J-factor model We tested 9 different configurations 12 / 26

13 Credit risk models Correlation modelling Hoeffding and risk decomposition Specific-systematic decomposition of the loss Because we deal with the trading book, this distinction between systemic and specific risks is key in the understanding of the number of factors on risk measure L(Z, ε) = k EAD k LGD k 1 {βk Z+ 1 β k β k ɛ k Φ 1 (p k )} Hoeffding decomposition of the default losses VAN DER VAART (2000) [20], ROSEN & SAUNDERS (2010) [14], HOEFFDING (1948) [21]. L(Z, ɛ) = E [L] } φ (L) : Expected Loss + E [L Z] E [L] } φ 1 (L; Z) : Systematic Loss + E [L ε] E [L] } φ 2 (L; ε) : Specific Loss + L(Z, ɛ) E [L Z] E [L ε] + E [L] } φ 1,2 (L; Z, ε) : Interaction Loss - φ 1 (L; Z) corresponds (up to the expected loss term) to the heterogeneous Large Pool Approximation (E [L Z] complete for infinitely fine grained, portfolio invariance property) 13 / 26

14 Portfolio risk and contributions Credit risk models Correlation modelling Hoeffding and risk decomposition Portfolio risk - Value-at-Risk (one-year 99.9%): VaR α[l] = inf{l R P(L l) α} - Full allocation property: VaR α[l = L 1 +L 2 ] = E [L 1 L = VaR α[l]]+e [L 2 L = VaR α[l]] Systematic-specific contribution of the portfolio risk VaR α[l] = E [φ L = VaR α[l]] } C : Expected Loss Contribution + E [φ 1 (L; Z) L = VaR α[l]] } C 1 (L; Z) : Systematic Contribution + E [φ 2 (L; ε) L = VaR α[l]] } C 2 (L; ε) : Specific Contribution + E [φ 1,2 (L; Z, ε) L = VaR α[l]] } C 1,2 (L; Z, ε) : Interaction Contribution Remark Factor contributions really depend on the projection made, while the risk remains the same 14 / 26

15 Portfolios - itraxx Europe - Corporates EU Corporate exposures: long only and long/short portfolios Impact on 99.9% VaR Drivers of risk (systematic vs idiosyncratic) A diversification portfolio and a hedge portfolio are built This parallels the distinction between the banking book (long positions, e.g. loans) and the trading book (long/short positions, e.g. in bonds, CDSs) 15 / 26

16 1-year Default Probabilities EU Corporate exposures: long only and long/short portfolios Impact on 99.9% VaR Drivers of risk (systematic vs idiosyncratic) 1-year Default Probabilities: Bloomberg Issuer Default Risk Methodology is used 16 / 26

17 Correlation matrices - Distributions EU Corporate exposures: long only and long/short portfolios Impact on 99.9% VaR Drivers of risk (systematic vs idiosyncratic) 17 / 26

18 Impacts on the risk - Long portfolio EU Corporate exposures: long only and long/short portfolios Impact on 99.9% VaR Drivers of risk (systematic vs idiosyncratic) 18 / 26

19 Impacts on the risk - Long-short portfolio EU Corporate exposures: long only and long/short portfolios Impact on 99.9% VaR Drivers of risk (systematic vs idiosyncratic) 19 / 26

20 EU Corporate exposures: long only and long/short portfolios Impact on 99.9% VaR Drivers of risk (systematic vs idiosyncratic) Systematic contribution to the risk - Long portfolio 20 / 26

21 EU Corporate exposures: long only and long/short portfolios Impact on 99.9% VaR Drivers of risk (systematic vs idiosyncratic) Systematic contribution to the risk - Long-short portfolio 21 / 26

22 EU Corporate exposures: long only and long/short portfolios Impact on 99.9% VaR Drivers of risk (systematic vs idiosyncratic) Conclusions - RWA variability and comparability The RWA variability stemming from correlation modelling remains high despite the correlation constraints - It is a challenge regarding model comparability. - Two-factor constraint is more suitable during stressed periods (2008) - The prescriptions might prove quite useful when dealing with a large number of assets: unconstrained correlation matrix (with small eigenvalues) would ease the building of opportunistic portfolios. Other main sources of variability are revealed - The high confidence level of the regulatory risk measure; - Disparities among correlation matrices (type of data and/or the calibration period). Small changes in exposures or other parameters may lead to significant changes in the credit VaR, jeopardizing the comparability of RWA. The use of Large Pool Approximation is questionable: poor contribution to the VaR of the systematic risk, for the trading book Bending the binds does not seem fundamental enough yet Need for more (regulatory) research on impacts on regulatory risk of estimation and calibration methods of the correlation matrix 22 / 26

23 Bibliography I EU Corporate exposures: long only and long/short portfolios Impact on 99.9% VaR Drivers of risk (systematic vs idiosyncratic) A. G. Haldane, Constraining discretion in bank regulation, Available at: BCBS, Revisions to the Basel II market risk framework, Available from: BCBS, Regulatory consistency assessment program (RCAP) - Analysis of risk-weighted assets for market risk, Available from: BCBS, Regulatory consistency assessment program (RCAP) - Second report on risk-weighted assets for market risk in the trading book, Available from: BCBS, Analysis of the trading book hypothetical portfolio exercise, Available from: K. Düllmann, M. Scheicher, and C. Schmieder, Asset correlations and credit portfolio risk: An empirical analysis, tech. rep., Discussion Paper, Series 2: Banking and Financial Supervision, BCBS, Fundamental review of the trading book (consultative paper 1), Available from: BCBS, Fundamental review of the trading book: A revised market risk framework (consultative paper 2), Available from: / 26

24 Bibliography II EU Corporate exposures: long only and long/short portfolios Impact on 99.9% VaR Drivers of risk (systematic vs idiosyncratic) BCBS, Fundamental review of the trading book: Outstanding issues (consultative paper 3), Available from: J.-P. Laurent, M. Sestier, and S. Thomas, Trading book and credit risk : how fundamental is the basel review?, Working paper - submitted to The Journal of Banking and Finance, Available from: id= , S. Wilkens and M. Predescu, Incremental default risk (idr): Modeling framework for the basel 4 risk measure, Available at SSRN , R. C. Merton, On the pricing of corporate debt: The risk structure of interest rates, The Journal of Finance, vol. 29, no. 2, pp , B. Committee, International Convergence of Capital Measurement and Capital Standards: A Revised Framework, Tech. Rep. November, D. Rosen and D. Saunders, Risk Factor Contributions in Portfolio Credit Risk Models, Journal of Banking & Finance, vol. 34, pp , Feb B. C. on Banking Supervision and B. for International Settlements, Instruction for basel III monitoring, BCBS, Frequently asked questions: Impact study on the proposed frameworks for market risk and CVA risk, Available from: impact study.pdf, / 26

25 Bibliography III EU Corporate exposures: long only and long/short portfolios Impact on 99.9% VaR Drivers of risk (systematic vs idiosyncratic) J. Andersen, L. Sidenius and S. Basu, All your hedge in one basket, Risk, pp , E. G. Birgin, J. M. Martínez, and M. Raydan, Nonmonotone spectral projected gradient methods on convex sets, SIAM Journal on Optimization, vol. 10, no. 4, pp , E. G. Birgin, J. M. Martínez, and M. Raydan, Algorithm 813: Spg software for convex-constrained optimization, ACM Transactions on Mathematical Software (TOMS), vol. 27, no. 3, pp , A. W. Van der Vaart, Asymptotic statistics, vol. 3. Cambridge university press, W. Hoeffding, A class of statistics with asymptotically normal distribution, The Annals of Mathematical Statistics, vol. 19, no. 3, pp , / 26

26 Disclaimer Basel recommandations on credit risk EU Corporate exposures: long only and long/short portfolios Impact on 99.9% VaR Drivers of risk (systematic vs idiosyncratic) Jean-Paul Laurent acknowledges support from the BNP Paribas Cardif chair Management de la Modélisation. Michael Sestier and Stéphane Thomas acknowledge support from PHAST Solutions Group. The usual disclaimer applies. Jean-Paul Laurent is Professor of Finance at the University Paris-1 Panthéon-Sorbonne(PRISM laboratory) and Member of the Labex ReFi. Michael Sestier is PhD Candidate at the University Paris-1 Pantheon-Sorbonne (PRISM laboratory) and Financial Engineer at PHAST Solutions Group. Stéphane Thomas PhD in Finance, is Managing Partner at PHAST Solutions Group and Member of the Labex ReFi. This work was achieved through the Laboratory of Excellence on Financial Regulation (Labex ReFi) supported by PRES hesam under the reference ANR-10-LABX It benefited from a French government support managed by the National Research Agency (ANR) within the project Investissements d Avenir Paris Nouveaux Mondes (investments for the future Paris-New Worlds) under the reference ANR-11-IDEX / 26