Recent Enhancements - PowerPoint Presentation

Slide1

Recent Enhancements Towards ConsistentCredit Risk Modelling Across Risk Measures

Disclaimer: The contents of this presentation are for discussion purposes only, representthe presenter’s views only and are not intended to represent the opinions of any firm orinstitution. None of the methods described herein is claimed to be in actual use.

RISK – Quant Congress USA16-18 July 2014, New York

Péter

DobránszkySlide2

IntroductionWe investigate in this presentation the link between credit spread and rating migration evolutionsThe first building block of a consistent modelling framework is the construction of appropriate generic credit spread curvesThe 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 curvesDifficulties with the intersection method – be granular but also robust and stableStatic representation vs. capturing the dynamicsCapturing 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 probabilitiesDeal with observed autocorrelation2Slide3

AssumptionsWe assume that we have a large enough history of spreads and corresponding ratings , where denotes a calendar date and identifies an issuer.We do not model the spread dynamics directly, but rather the log-spread dynamics. Accordingly, for convenience, we deal with

in the following equations.These log-spreadsIf the data is not clean enough we may assign a weight for each issuer and for each day. 3Slide4

Least-square regressionIn 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.4Slide5

AgendaCDS curvesCapturing dynamics – sectorial approach (VaR, CVA VaR)Factor analysis, Random Matrix Theory, ClusteringEstimating level – proxy spread curves (CVA, CS01, SEEPE)GroupingRating migration effect (EEPE)Migration matrixEstimation error (IRC, CRM)Default probabilities and recovery rateSovereigns

(IRC, CRM)Risk premiumJoint default events and correlated migration moves modellingConcentration of events (IRC, CRM)Some double counting issues5Slide6

CDS curvesUnderstand the business cycles - stochastic business time (GARCH, etc.)Detach business time (by sectors) from calendar timeVaR vs. Stressed VaR, EEPE vs. Stressed EEPE 6Slide7

CDS curvesExpected value of integrated business time over a calendar time periodDynamics of ATM implied volatility for various maturitiesCredit spreads as annual average default ratesCorrelated, but standalone clocks7Slide8

CDS curvesWhat is stationary?Log-returns (?)8

Kurtosis by return typeRaw absoluteNormalised absoluteRaw relativeNormalised relativeiTraxx Eur

5Y12.710.77.25.3

CDX.NA.IG 5Y

13.7

6.5

4.9

3.7Slide9

CDS curvesCapture the dynamics of spreads (VaR, CVA VaR, CRM)Merge, demerge, new namesIlliquid curves – systemic, sectorial, idiosyncratic risk componentsSelection of liquid curves as basis for capturing the dynamicsDefinition of liquidity – contributors, number of non-updatesSectorial approach – mapping of names to groupsGroups of names with similarities, homogeneityLarge enough and small enough groups, concentrationTrade-off between specificity and calibration uncertaintyRepresentation by number of names and by exposuresCan you assume

cross-sectional relationships?(N industry + M sector) systemic factors(N industry × M sector) systemic factors9Slide10

CDS curvesPrincipal Component Analysis of CDS log-returns, decompose correlationBiased figures if not accounted for stochastic business timesAssume a group of 10 with an extra 30% variance explanation within groupThis specific group factor explains only of total variance

 10Slide11

CDS curvesAssume 10 explanatory factors and remove their impactDoes 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 group11Slide12

CDS curvesRandom Matrix Theory: it explains the eigenvalue distributionIf , with a fixed ratio , the eigenvalue spectral density of the correlation matrix is given by

where

.But it works also for . For instance,

.

12Slide13

CDS curvesPrincipal components are often misleadingIf you have risk factors and you remove the impact of the first principal components, then the remaining variance is explained by only instead of factors.Therefore, the first principal components explains also part of the noise variance.Assume a one-factor model for 4 variables with pairwise correlations of 50%Remove the impact of the first principal component – remaining parts of the 4 variables are not uncorrelated, instead, their correlation is -33%! – over-compensationRemove the mean – mean captures only part of the systemic factor’s variance, thus

the remaining parts of the 4 variables have a 6% correlation – under-compensation 13100%-33%-33%-33%-33%

100%-33%-33%-33%-33%

100%

-33%

-33%

-33%

-33%

100%

100%

6%

6%

6%

6%

100%

6%

6%

6%

6%

100%

6%

6%

6%

6%

100%Slide14

CDS curvesClusteringIt is a technique that collects together series of values into groups that exhibit similar behaviour.Hierarchical clustering based on Euclidean distance or correlation14

Still mapping of clusters to sectors and regions are requiredNot robust towards outliers, few small clusters and large concentrationLarge clusters should be re-clustered

Does not ensure homogeneity within cluster – fixed number of clustersRecent: Make 2 clusters, split each cluster into 2 until RMT conditions are met – still exposed to outliersSlide15

CDS curvesEstimating level for a given day – proxy spread curves (CVA, CS01, SEEPE)Data mining like explorationHow many distinguishable groups are there? Split by how many dimensions?Basel III requests split by sectors, regions and ratings (see EBA BTS)15Slide16

CDS curvesHypothesis test: difference between meansApply 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

16Slide17

CDS curvesTwo-sample t-test – compare groups based on two-tailed testsNull hypothesis: considering two sectors, their average spread levels are equal. Alternative hypothesis: average spread levels are not equal, thus these sectors require separate proxy spread curves.Assume that the standard deviations by samples are different. Therefore, compute the standard error (SE) of the sampling distribution as

The distribution of the statistic can be closely approximated by the t distribution with degrees of freedom (DF) calculated as

The

test statistic

is

a t-score (t) defined by

.

17Slide18

CDS curvesP-valuesGrouping may change as sector levels fluctuateDefines minimum number of names in a groupHere 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 2008P-values of the two-sample t-test as of 15 June 2012Slide19

CDS curves19As of 15 June 2012As of 31 December 2008

Useful cross-sectional informationRecently slope is not 1

 Slide20

CDS curvesRating dependency for various sectors and regionsDifferent slope coefficients may be requiredBigger difference between sectors than regions20Slide21

CDS curvesRating 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;Slide22

Migration matrixEstimation error (IRC, CRM)Less or more rating matrices? Trade-off between capturing better the specific risk profiles and basic risk vs. reducing the estimation noise.Which ones? Sovereign and corporate migration matrices? Corporate divided by region (US / Europe) and industry (financial / non-financial)?Relevance for bank portfolio vs. availability of data, i.e. available data often with US concentration.Finer rating grid may reduce the jump of P&Ls on the tails, but it introduces estimation noise.Binomial proportion confidence interval, i.e. how reliable is the transition probability estimate

?CLT:, Wilson interval:

For a 95% confidence interval:

Enormous IRC impact

Smoothing?

22

Trial

Outcome

Estimate

Lower CI

Upper CI

50

1

2.0%

0.4%

10.5%

100

1

1.0%

0.2%

5.4%

500

1

0.2%

0.0%

1.1%

1000

1

0.1%

0.0%

0.6%Slide23

Migration matrixCalculation of short-term transition matricesMarkov approach: Assume time-homogeneous continuous-time Markov chain and scale the transition matrix via the generator matrix.Which is the best short-term matrix which provides that multiplying it by itself several times gives the best approximation for the original one-year matrix?

=

Cohort

method

: Discrete-time method based on the historical migration data. Calibrations to various time horizons may show autocorrelation in migrations.

Maximum likelihood estimation assuming Markov model

Accounting for stochastic business times

23Slide24

Default probabilities and recovery ratesSource of recovery ratesWhat are the local currency recovery rates?Sovereigns may go default on their hard currency and local currency obligations separatelyDoes the IRC engine simulate both events, if yes, how to manage correlation, if not, which rating is used for IRC calculationsIt can be interpreted as what is the LC/HC bond value in case the HC/LC bond migrate or defaultVarious approaches to adjust the LC recovery rates to account for FX depreciation – quanto CDSs may be usedWhat 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 rate24Slide25

Default probabilities and recovery ratesSource 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.25AAA

1Y2Y3Y4Y5Y7Y10YPhysical0.00%0.00%

0.00%0.00%0.00%0.00%0.01%Risk Neutral

0.18%

0.23%

0.29%

0.35%

0.40%

0.43%

0.45%

AA

1Y

2Y

3Y

4Y

5Y

7Y

10Y

Physical

0.00%

0.01%

0.01%

0.01%

0.01%

0.02%

0.02%

Risk Neutral

0.28%

0.35%

0.43%

0.52%

0.60%

0.65%

0.71%

A

1Y

2Y

3Y

4Y

5Y

7Y

10Y

Physical

0.02%

0.02%

0.03%

0.04%

0.05%

0.07%

0.10%

Risk Neutral

0.36%

0.45%

0.55%

0.65%

0.74%

0.80%

0.88%

BBB

1Y

2Y

3Y

4Y

5Y

7Y

10Y

Physical

0.12%

0.15%

0.19%

0.22%

0.25%

0.31%

0.38%

Risk Neutral

0.53%

0.68%

0.82%

0.96%

1.10%

1.19%

1.30%

BB

1Y

2Y

3Y

4Y

5Y

7Y

10Y

Physical

0.74%

0.84%

0.94%

1.02%

1.10%

1.21%

1.32%

Risk Neutral

1.20%

1.59%

1.94%

2.24%

2.45%2.60%2.70%        B1Y2Y3Y4Y5Y7Y10YPhysical3.33%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%5.62%        CCC1Y2Y3Y4Y5Y7Y10YPhysical12.70%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.

T

aking

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

.Slide26

Default probabilities and recovery ratesAccounting for risk premiumBanks take over risk, diversify and get compensation for systemic riskDiffusion processes: risk premium over risk is negligible in the short-termRisk premium related to jump risk and gap risk is priced differentlyBB sector 5Y CDS ranged between 100 and 700 bps from beginning 2006 to mid 2011Implied default rate around 1.7-11% () vs.

TTC default rate of 1%Rare events (AAA) are priced with higher risk premiumProblems started to rise with Basel 2.5IRC loss distribution is strongly effected by risk premium

Visualise the potential time value effect when risk premium is significant

26Slide27

Default probabilities and recovery ratesShort protection portfolio of CDSs written on BB rated issuers30th June 2009Average 1Y CDS spread of the constituents was 600 bpsIn 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 realised27Slide28

Joint default events and correlated migration moves

Asset value correlation: parameter of the Gaussian copula approachDefault correlation (Pearson correlation):If CEDFj is not equal to CEDFk, the default correlation can never reach 100%Process correlation: when processes are moving together28

Time fractions of co-movements

T

T+∆t

j

kSlide29

Gaussian casePairwise correlations determine the whole joint dependence structureProxies for calibrationFactor correlation approach (KMV GCorr)Same correlation for defaults and migrationsCopula: one-step discrete-time approachForward joint density does not existAsset value correlation model

29Slide30

Term structure of default correlationsFix 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 ratingsThe lower the cumulative probability of defaults the lower is the default correlationMost copula based approaches imply that the defaults of highly rated names are basically independentOpposite to this, process correlations produce flat default correlation curves

30Slide31

Default correlations by rating classesGaussian copulaAVC = 10%Correlated continuous-time Markov chainsProcess corr. = 11%Time fraction 1.2%

31

AAAAAABBB

BB

B

CCC

AAA

0.00%

0.01%

0.01%

0.03%

0.05%

0.07%

0.08%

AA

0.01%

0.03%

0.04%

0.08%

0.15%

0.23%

0.28%

A

0.01%

0.04%

0.06%

0.12%

0.23%

0.35%

0.44%

BBB

0.03%

0.08%

0.12%

0.28%

0.53%

0.82%

1.05%

BB

0.05%

0.15%

0.23%

0.53%

1.043%

1.65%

2.17%

B

0.07%

0.23%

0.35%

0.82%

1.65%

2.66%

3.60%

CCC

0.08%

0.28%

0.44%

1.05%

2.17%

3.60%

5.03%

AAA

AA

A

BBB

BB

B

CCC

AAA

0.02%

0.06%

0.05%

0.04%

0.03%

0.02%

0.01%

AA

0.06%

1.08%

0.68%

0.27%

0.10%

0.05%

0.03%

A

0.05%

0.68%

0.73%

0.39%

0.19%

0.11%

0.05%

BBB

0.04%

0.27%

0.39%

0.86%

0.53%

0.29%

0.15%

BB

0.03%

0.10%

0.19%

0.53%

1.043%

0.59%

0.33%

B

0.02%

0.05%

0.11%

0.29%

0.59%

1.14%

0.65%

CCC

0.01%

0.03%

0.05%

0.15%

0.33%

0.65%

1.16%

Moody's

KMV

analysis

Realised

default correlation

A1-A3

0.65%

Baa1-Baa3

0.59%

Ba1-Ba3

1.68%B1-B3 & below2.36%

different default correlation structure by rating

term structure is flat at PC

2

if T is smallSlide32

Event concentration – the new dimension of uncertaintyIn case of jumpy processes the parameterisation of the pairwise dependence structures is not enough to determine the N-joint lawSame pairwise dependence structure, but different N-joint lawHigh concentration: Armageddon scenario likelyLow concentration: probability of large number of defaults is high

32

High concentration

Low concentration

j

k

l

j

k

l

T+∆t

T+∆t

T

TSlide33

Incremental modelling uncertaintyCompare Gaussian copula model against more advanced correlated jump models with various event concentrationsAVC = 8.5%, which means process correlation of 10%PD, LGD and P&L effect of rating changes are the same in each caseFixed time horizon of one year

In case of small portfolios, various models produce very similar IRC loss distributions33Slide34

Incremental modelling uncertaintyThe larger the portfolio the larger the impact of the model choiceEspecially short protection portfolios are very sensitive to the concentration modelling – concentration of default events can hardly be calibrated

IRCAVCPDLPDMPDHBB long9.1 M7.6 M7.9 M7.4 MBB short3.4 M0.4 M4.2 M5.3 M

34Slide35

Separating default and migration correlationsUntil 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 correlationsCritics for reduced-form models correlating only default intensities

35

AAAAAA

BBB

BB

B

CCC

AAA

0.54%

0.26%

0.39%

0.26%

0.11%

0.07%

0.00%

AA

0.26%

0.23%

0.42%

0.23%

0.08%

0.04%

0.00%

A

0.39%

0.42%

1.30%

0.97%

0.44%

0.22%

0.01%

BBB

0.26%

0.23%

0.97%

1.90%

0.89%

0.54%

0.04%

BB

0.11%

0.08%

0.44%

0.89%

0.775%

0.58%

0.09%

B

0.07%

0.04%

0.22%

0.54%

0.58%

0.80%

0.15%

CCC

0.00%

0.00%

0.01%

0.04%

0.09%

0.15%

0.32%

Moody's

KMV

analysis

Realised

default correlation

A1-A3

0.65%

Baa1-Baa3

0.59%

Ba1-Ba3

1.68%

B1-B3 & below

2.36%Slide36

Stochastic business time36

Time homogeneity is clearly not an appropriate assumptionStress periods are described by volatility clustersSlide37

Stochastic business timeRecent time changed models are designed to explain default correlationsUse a realistic statistical model to describe the business time dynamics

37Slide38

Stochastic business timeCalibrate the transition generator by assuming stochastic business timeWhat 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-yearAAAAAA

BBB

BB

B

CCC

AAA

0.00%

0.00%

0.00%

0.01%

0.01%

0.03%

0.04%

AA

0.00%

0.00%

0.01%

0.01%

0.03%

0.05%

0.08%

A

0.00%

0.01%

0.01%

0.03%

0.06%

0.11%

0.18%

BBB

0.01%

0.01%

0.03%

0.07%

0.15%

0.28%

0.46%

BB

0.01%

0.03%

0.06%

0.15%

0.324%

0.60%

0.99%

B

0.03%

0.05%

0.11%

0.28%

0.60%

1.13%

1.86%

CCC

0.04%

0.08%

0.18%

0.46%

0.99%

1.86%

3.09%

Moody's

KMV

analysis

Realised

default correlation

A1-A3

0.65%

Baa1-Baa3

0.59%

Ba1-Ba3

1.68%

B1-B3 & below

2.36%

AAA

AA

A

BBB

BB

B

CCC

1-day

0.01%

1.22%

1.22%

1.22%

1.222%

1.23%

1.25%

1-month

0.02%

1.21

%

1.13%

1.15%

1.224%

1.35%

1.79%

1-year

0.02%

1.04%

0.69%

0.92%

1.350%

2.26%

4.21%Slide39

Some double counting issuesConsistency and coherency issues between capital chargesPotential 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 nowSimilarly 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 questionableSudden and expected defaults shall be separated and capitalised accordingly 39