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Recent Enhancements Towards Consistent Credit Risk Modelling Across Risk Measures Disclaimer: The contents of this presentation are for discussion purposes only, represent the presenter’s views only and are not intended to represent the opinions of any firm or institution. None of the methods described herein is claimed to be in actual use. RISK – Quant Congress USA 16-18 July 2014, New York Péter Dobránszky

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Introduction We investigate in this presentation the link between credit spread and rating migration evolutions The first building block of a consistent modelling framework is the construction of appropriate generic credit spread curves The main applications are VaR, IRC, CRM, CCR, CVA, etc. Regulatory requirements for Regulatory CVA, credit VaR, wrong-way risk modelling, economic downturn modelling, incremental default in FRTB, etc. We present a fully cross-sectional approach for building generic credit spread curves aka proxy spread curves Difficulties with the intersection method – be granular but also robust and stable Static representation vs. capturing the dynamics Capturing times of stress and benign periods, stochastic business time, mean- reversions, regime switching modelling, etc. Take into account risk premium, jump risk, gap risk, historical vs. risk neutral probabilities Deal with observed autocorrelation 2

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Assumptions 3

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Least-square regression In the course of calibrating the rating distances we disregard the potential sector and region dimensions of the spreads and we will assume that the rating distances are static. Accordingly, we intend to calibrate the rating distances by the following regression. 4

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Agenda CDS curves Capturing dynamics – sectorial approach (VaR, CVA VaR) Factor analysis, Random Matrix Theory, Clustering Estimating level – proxy spread curves (CVA, CS01, SEEPE) Grouping Rating migration effect (EEPE) Migration matrix Estimation error (IRC, CRM) Default probabilities and recovery rate Sovereigns (IRC, CRM) Risk premium Joint default events and correlated migration moves modelling Concentration of events (IRC, CRM) Some double counting issues 5

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CDS curves Understand the business cycles - stochastic business time (GARCH, etc.) Detach business time (by sectors) from calendar time VaR vs. Stressed VaR, EEPE vs. Stressed EEPE 6

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CDS curves Expected value of integrated business time over a calendar time period Dynamics of ATM implied volatility for various maturities Credit spreads as annual average default rates Correlated, but standalone clocks 7

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CDS curves What is stationary? Log-returns (?) 8 Kurtosis by return typeRaw absoluteNormalised absoluteRaw relativeNormalised relative iTraxx Eur 5Y12.710.77.25.3 CDX.NA.IG 5Y13.76.54.93.7

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CDS curves Capture the dynamics of spreads (VaR, CVA VaR, CRM) Merge, demerge, new names Illiquid curves – systemic, sectorial, idiosyncratic risk components Selection of liquid curves as basis for capturing the dynamics Definition of liquidity – contributors, number of non-updates Sectorial approach – mapping of names to groups Groups of names with similarities, homogeneity Large enough and small enough groups, concentration Trade-off between specificity and calibration uncertainty Representation by number of names and by exposures Can you assume cross-sectional relationships? (N industry + M sector) systemic factors (N industry × M sector) systemic factors 9

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CDS curves 10

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CDS curves Assume 10 explanatory factors and remove their impact Does the remaining part behaves like random independent noise? Still there can be 50 groups of 10 names with an extra group factor explaining 30% of the variance within the group 11

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CDS curves 12

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CDS curves 13

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CDS curves Clustering It is a technique that collects together series of values into groups that exhibit similar behaviour. Hierarchical clustering based on Euclidean distance or correlation 14 Still mapping of clusters to sectors and regions are required Not robust towards outliers, few small clusters and large concentration Large clusters should be re-clustered Does not ensure homogeneity within cluster – fixed number of clusters Recent: Make 2 clusters, split each cluster into 2 until RMT conditions are met – still exposed to outliers

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CDS curves Estimating level for a given day – proxy spread curves (CVA, CS01, SEEPE) Data mining like exploration How many distinguishable groups are there? Split by how many dimensions? Basel III requests split by sectors, regions and ratings (see EBA BTS) 15

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CDS curves Hypothesis test: difference between means Apply the so-called two-sample t-test, which is appropriate when the following conditions are met: The sampling method for each sample is simple random sampling. The samples are independent. Each sample is drawn from a normal or near-normal population. The first two conditions are met by construction. Concerning the third rule, by rules-of-thumb, a sampling distribution is considered near-normal if any of the following conditions apply: The sample data are symmetric, unimodal, without outliers, and the sample size is 15 or less. The sample data are slightly skewed, unimodal, without outliers, and the sample size is 16 to 40. The sample size is greater than 40, without outliers. Analyse log-spreads and normalise by the rating effect 16

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CDS curves 17

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CDS curves P-values Grouping may change as sector levels fluctuate Defines minimum number of names in a group Here only European issuers, however, is it the same in NA? Are there cross-sectional information being useful? 18 P-values of the two-sample t-test as of 31 December 2008 P-values of the two-sample t-test as of 15 June 2012

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CDS curves 19 As of 15 June 2012 As of 31 December 2008 Useful cross-sectional information Recently slope is not 1

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CDS curves Rating dependency for various sectors and regions Different slope coefficients may be required Bigger difference between sectors than regions 20

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CDS curves Rating migration effect (EEPE) BIS Quarterly Review, June 2004: “Rating announcements affect spreads on credit default swaps. The impact is more pronounced for negative reviews and downgrades than for outlook changes.” 21 Regulation, CRR, Article 158: (i) for institutions using the Internal Model Method set out in Section 6 of Chapter 6, to calculate the exposure values and having an internal model permission for specific risk associated with traded debt positions in accordance with Part Three, Title IV, Chapter 5, M shall be set to 1 in the formula laid out in Article 148(1), provided that an institution can demonstrate to the competent authorities that its internal model for Specific risk associated with traded debt positions applied in Article 373 contains effects of rating migrations;

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Migration matrix 22 TrialOutcomeEstimateLower CIUpper CI 5012.0%0.4%10.5% 10011.0%0.2%5.4% 50010.2%0.0%1.1% 100010.1%0.0%0.6%

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Migration matrix 23

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Default probabilities and recovery rates Source of recovery rates What are the local currency recovery rates? Sovereigns may go default on their hard currency and local currency obligations separately Does the IRC engine simulate both events, if yes, how to manage correlation, if not, which rating is used for IRC calculations It can be interpreted as what is the LC/HC bond value in case the HC/LC bond migrate or default Various approaches to adjust the LC recovery rates to account for FX depreciation – quanto CDSs may be used What are the recovery rates for covered bonds and government guarantees? The rating of issuing bank is taken, which implies “high” PD, but when the issuer goes to default, there is still a pool of assets or another guarantor to meet the obligation. Recovery rates are usually high to compensate that “wrong” PDs are used. Ensure that bond PV < recovery rate 24

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Default probabilities and recovery rates Source of estimated or implied probabilities of defaults (PD) Historical TTC default probabilities provided by rating agencies (cohort). Risk-neutral PIT default probabilities bootstrapped from traded CDSs. 25 AAA1Y2Y3Y4Y5Y7Y10Y Physical0.00% 0.01% Risk Neutral0.18%0.23%0.29%0.35%0.40%0.43%0.45% AA1Y2Y3Y4Y5Y7Y10Y Physical0.00%0.01% 0.02% Risk Neutral0.28%0.35%0.43%0.52%0.60%0.65%0.71% A1Y2Y3Y4Y5Y7Y10Y Physical0.02% 0.03%0.04%0.05%0.07%0.10% Risk Neutral0.36%0.45%0.55%0.65%0.74%0.80%0.88% BBB1Y2Y3Y4Y5Y7Y10Y Physical0.12%0.15%0.19%0.22%0.25%0.31%0.38% Risk Neutral0.53%0.68%0.82%0.96%1.10%1.19%1.30% BB1Y2Y3Y4Y5Y7Y10Y Physical0.74%0.84%0.94%1.02%1.10%1.21%1.32% Risk Neutral1.20%1.59%1.94%2.24%2.45%2.60%2.70% B1Y2Y3Y4Y5Y7Y10Y Physical3.33%3.47%3.56%3.62%3.66%3.68%3.65% Risk Neutral2.81%3.57%4.30%4.94%5.50%5.62% CCC1Y2Y3Y4Y5Y7Y10Y Physical12.70%12.26%11.84%11.46%11.10%10.51%9.85% Risk Neutral5.87%7.15%8.05%8.72%9.00%8.80%8.48% Comparison of transformed historical PDs with Markit sector curves as of 30 June 2009 and assuming 40% recovery rate. Taking non-diversifiable risk is compensated by premium. The rarer the event the more difficult to diversify and the higher the risk premium. IRC: historical PDs are used for simulations, while implied default probabilities are used for re-pricing. Impact depends on the portfolio.

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Default probabilities and recovery rates 26

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Default probabilities and recovery rates Short protection portfolio of CDSs written on BB rated issuers 30 th June 2009 Average 1Y CDS spread of the constituents was 600 bps In case no default or migration event happens, expected portfolio P&L is around 6% Not accounting for time value, expected portfolio P&L is around -1% (TTC) Numerous default events may occur before any effective loss is realised 27

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Joint default events and correlated migration moves Asset value correlation: parameter of the Gaussian copula approach Default correlation (Pearson correlation): If CEDF j is not equal to CEDF k, the default correlation can never reach 100% Process correlation: when processes are moving together 28 Time fractions of co-movements TT+∆t j k

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Gaussian case Pairwise correlations determine the whole joint dependence structure Proxies for calibration Factor correlation approach (KMV GCorr) Same correlation for defaults and migrations Copula: one-step discrete-time approach Forward joint density does not exist Asset value correlation model 29

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Term structure of default correlations Fix the AVC and measure the Pearson default correlation for various horizons (annual PD = 2%, 2-state Markov chain with jump-to-default) Similar term structure of default correlations by ratings The lower the cumulative probability of defaults the lower is the default correlation Most copula based approaches imply that the defaults of highly rated names are basically independent Opposite to this, process correlations produce flat default correlation curves 30

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Default correlations by rating classes Gaussian copula AVC = 10% Correlated continuous-time Markov chains Process corr. = 11% Time fraction 1.2% 31 AAAAAABBBBBBCCC AAA0.00%0.01% 0.03%0.05%0.07%0.08% AA0.01%0.03%0.04%0.08%0.15%0.23%0.28% A0.01%0.04%0.06%0.12%0.23%0.35%0.44% BBB0.03%0.08%0.12%0.28%0.53%0.82%1.05% BB0.05%0.15%0.23%0.53%1.043%1.65%2.17% B0.07%0.23%0.35%0.82%1.65%2.66%3.60% CCC0.08%0.28%0.44%1.05%2.17%3.60%5.03% AAAAAABBBBBBCCC AAA0.02%0.06%0.05%0.04%0.03%0.02%0.01% AA0.06%1.08%0.68%0.27%0.10%0.05%0.03% A0.05%0.68%0.73%0.39%0.19%0.11%0.05% BBB0.04%0.27%0.39%0.86%0.53%0.29%0.15% BB0.03%0.10%0.19%0.53%1.043%0.59%0.33% B0.02%0.05%0.11%0.29%0.59%1.14%0.65% CCC0.01%0.03%0.05%0.15%0.33%0.65%1.16% Moody's KMV analysis Realised default correlation A1-A30.65% Baa1-Baa30.59% Ba1-Ba31.68% B1-B3 & below2.36% different default correlation structure by rating term structure is flat at PC 2 if T is small

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Event concentration – the new dimension of uncertainty In case of jumpy processes the parameterisation of the pairwise dependence structures is not enough to determine the N-joint law Same pairwise dependence structure, but different N-joint law High concentration: Armageddon scenario likely Low concentration: probability of large number of defaults is high 32 High concentrationLow concentration j k l j k l T+∆t TT

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Incremental modelling uncertainty Compare Gaussian copula model against more advanced correlated jump models with various event concentrations AVC = 8.5%, which means process correlation of 10% PD, LGD and P&L effect of rating changes are the same in each case Fixed time horizon of one year In case of small portfolios, various models produce very similar IRC loss distributions 33

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Incremental modelling uncertainty The larger the portfolio the larger the impact of the model choice Especially short protection portfolios are very sensitive to the concentration modelling – concentration of default events can hardly be calibrated IRCAVCPDLPDMPDH BB long9.1 M7.6 M7.9 M7.4 M BB short3.4 M0.4 M4.2 M5.3 M 34

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Separating default and migration correlations Until this point we assumed the same correlation between default events and migration moves. Nevertheless, we can separate the Markov generator matrix for defaults and migrations. Even perfectly correlated migration moves cannot reproduce the realised default correlations Critics for reduced-form models correlating only default intensities 35 AAAAAABBBBBBCCC AAA0.54%0.26%0.39%0.26%0.11%0.07%0.00% AA0.26%0.23%0.42%0.23%0.08%0.04%0.00% A0.39%0.42%1.30%0.97%0.44%0.22%0.01% BBB0.26%0.23%0.97%1.90%0.89%0.54%0.04% BB0.11%0.08%0.44%0.89%0.775%0.58%0.09% B0.07%0.04%0.22%0.54%0.58%0.80%0.15% CCC0.00% 0.01%0.04%0.09%0.15%0.32% Moody's KMV analysis Realised default correlation A1-A30.65% Baa1-Baa30.59% Ba1-Ba31.68% B1-B3 & below2.36%

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Stochastic business time 36 Time homogeneity is clearly not an appropriate assumption Stress periods are described by volatility clusters

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Stochastic business time Recent time changed models are designed to explain default correlations Use a realistic statistical model to describe the business time dynamics 37

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Stochastic business time Calibrate the transition generator by assuming stochastic business time What degree of realised default correlation can be explained by SBT? Similarity with correlated default intensities (correlated migration only) Default correlation by rating is not flat! Combine with process correlation! Term structure of default correlation by PC = 11% plus SBT (hockey stick): 38 1-yearAAAAAABBBBBBCCC AAA0.00% 0.01% 0.03%0.04% AA0.00% 0.01% 0.03%0.05%0.08% A0.00%0.01% 0.03%0.06%0.11%0.18% BBB0.01% 0.03%0.07%0.15%0.28%0.46% BB0.01%0.03%0.06%0.15%0.324%0.60%0.99% B0.03%0.05%0.11%0.28%0.60%1.13%1.86% CCC0.04%0.08%0.18%0.46%0.99%1.86%3.09% Moody's KMV analysis Realised default correlation A1-A30.65% Baa1-Baa30.59% Ba1-Ba31.68% B1-B3 & below2.36% AAAAAABBBBBBCCC 1-day0.01%1.22% 1.222%1.23%1.25% 1-month0.02%1.21%1.13%1.15%1.224%1.35%1.79% 1-year0.02%1.04%0.69%0.92%1.350%2.26%4.21%

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Some double counting issues Consistency and coherency issues between capital charges Potential exposure within a year does not capture that losses in case of a future default have potentially been realised already by CVA VaR when spreads were climbing up – this CVA variation is capitalised now Similarly for IRC vs. VaR – if being long credit for Greece, daily MtM losses were capitalised by VaR, while there was no further loss at the time of default, thus IRC capital charge was questionable Sudden and expected defaults shall be separated and capitalised accordingly 39

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