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A Multi-Period Gaussian Copula Model of Default Risk

October 29, 2007

Gary Dunn – UK Financial Services AuthorityCharles Monet – Coordinator of AIGTB Technical Working Group

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Presentation Outline

• Background and policy context• Description of multi-period model

– Focus on intra-period and inter-period correlation issues

• Study results for sample portfolios– Multi-period model versus single-period (annual)

model– Changing the capital horizon– Changing the liquidity horizon– Changing the PD, given fixed capital and liquidity

horizons

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Background

• Banks operating under a VaR-based capital requirement for market risk need to hold capital against default risk in the trading book which is incremental to risk captured within VaR.

• Requirement for Incremental Default Risk Capital (IDRC) introduced in paragraph 718(xcii and xciii) of Revised Accord

• No generally agreed methodology in the industry for measuring default risk in the trading book

• AIGTB developing guidelines for IDRC consistent with the high-level principles of paragraph 718(xcii and xciii)

• Key principle of AIGTB is reliance on firms’ internal models of incremental default risk

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Key Supervisory Standards

• Soundness standard consistent with A-IRB• 99.9% confidence interval over 1-year capital horizon

– Constant level of risk over capital horizon (not constant position)• Benefit of liquidity

– Liquidity horizon defined as time to sell or completely hedge an exposure in a stressed market environment

– Shorter liquidity horizon reduces effective PD because annualized PD over liquidity horizon is lower than the annual PD

– Shorter liquidity horizon also reduces inter-period correlation compared to single-period approach

• Benefit of intra-obligor and inter-obligor hedging• Impact of concentrations

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Objectives of This Study

• Extend results of previous paper1

– That paper relied on a single-period model.– Did not fully capture economics of rolling over exposures

• Build multi-period simulation model to capture economics of default risk in the trading book– Multiple liquidity horizons within capital horizon of one year– Rebalancing of positions to constant level of risk at beginning of

each period• Evaluate impact of different parameter choices

– Capital horizon– Liquidity horizon– PD

(1) Dunn, G., Gibson, M., Ikosi, G., Jones, J., Monet, C., and Sullivan, M., “Assessing Alternative Assumptions on Default Risk Capital in the Trading Book, Working Paper, December 2006.

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Multi-Period Gaussian Copula Model

• Start with single-period Gaussian copula model over initial liquidity horizon– Initial positions

• Based on the existing portfolio of exposures– PD over the liquidity horizon

• PDs over short time horizons based on Moody’s historical data

• Market-oriented approach (e.g., credit spreads) to measure short-term PDs would be an alternative

– Asset correlation• Asset correlation in initial period set to be consistent with

asset correlation over one year time horizon• See discussion on slide 9

– Compute default losses over the first liquidity horizon

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Multi-Period Gaussian Copula Model (cont.)

• Use single-period Gaussian copula model to compute default losses in each subsequent liquidity horizon– Positions: Assume rebalancing to constant level of risk

• Use initial portfolio positions– PD: Same as PD in the initial liquidity horizon– Asset correlation

• Same as asset correlation in the initial liquidity horizon• Also, include correlation between liquidity horizons

– Compute default losses over each liquidity horizon– Repeat for each of N liquidity horizons in the capital horizon

• At end of each simulation run– Sum default losses over the N liquidity horizons– Result is the number of default losses over the capital horizon

• At end of 20,000 to 50,000 simulation runs– Compute 99.9% downside of default losses over the capital horizon

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Correlation Structure in Multi-Period Model

• To model intra-period correlation, we assumed an AR-1 process for the systematic variable– Assumes that defaults do not materialize instantly with respect to

economic and financial shocks– The systematic shocks from one liquidity horizon persist according to the

strength of a parameter – The return for obligor i in period t is:

)1,0(,1

:

11

1

,22

,

NormalfactorsystematictheofprocessARtheoftCoefficien

ncorrelatioassetperiodIntra

Where

x

itt

titti

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Correlation Structure in Multi-Period Model

• Given the AR-1 process on the prior page, the following relationship exists among the correlations if the annual asset correlation is to be preserved:

factorsystematictheofprocessARtheoftCoefficienhorizoncapitaltheinhorizonsliquidityofNumberL

LL

A

CorrCorrACorrCorr

L

AnnualAnnual

AnnualHorizonLiquidity

)1(

1)1(1

)1(21

)1(1

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Inter-Period Correlation: Alternative Assumptions• Simplest assumption is that =0

– Then, A=1 and CorrLiquidity Horizon=CorrAnnual

– Convenient and consistent with financial theory (Martingale)• Alternative: evaluate default correlations across periods

– Default losses are correlated across time periods due to the lagged impact of economic and financial innovations.

– ρ can be estimated based on default clustering across periods• Issues

– Given portfolio rebalancing in a trading context, does the impact of repricing reduce the effective inter-period correlation to zero in a rebalanced portfolio?

– How material is the assumption re: inter-period correlation?

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Impact of Inter-Period Correlation

Portfolio – See appendix =0 =0.25 =0.5 =0.75 =0.999

BB Long-Only 5.7% 4.6% 4.6% 4.6% 4.6%

BB Long-Short 4.6% 4.6% 4.6% 4.6% 4.6%

Long-Bias Test Portfolio 4.4% 4.4% 4.4% 4.4% 4.4%

Note: The abrupt change is due to the integer nature of the defaults. For example, the transition from 5.7%% to 4.6% for the BB Long-Only portfolio represents a change from 5 defaults to 4 defaults.

The following table shows the capital requirement as a percent of the gross long LGDs for each portfolio, assuming a 1-month liquidity horizon and a 1-year capital horizon.

• PD at the 1-month horizon is based on Moody’s data.• Annual asset correlation is 0.2.• is the coefficient of the inter-period AR-1 process of the

systematic credit variable.

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Impact of Inter-Period Correlation

• Inter-period correlation has almost no impact on the capital requirement– Not a surprise, because the annual asset correlation is being

preserved by applying the equation on page 9– There is a slight trend for the capital requirement to decline as

the inter-period correlation increases.• Impact is visible only in BB Long-Only portfolio• Cause is uncertain

• Therefore, the results in this paper are based on an inter-period correlation of zero– Does not indicate that we believe that the inter-period correlation

is zero– Convenient and simple

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Capital Comparisons

Goal: As far as possible, assess impact of different factors separately from changes in the other factors

1. Multi-period capital requirement versus single-period– Same total PD over the 1-year capital horizon

2. Changing the capital horizon– Total PD over the capital horizon necessarily changes– Also have impact of multiple liquidity horizons

3. Changing the liquidity horizon– Same total PD over the 1-year capital horizon

4. Changing the PD– Hold capital horizon and liquidity horizon constant

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1. Multi-Period Versus Single-Period Model

See Note 1 Single-period

Multi-period

Single ÷ Multi

Single-period

Multi-period

Single ÷ Multi

BBB Long-Only 3.2 2.5 1.3 5.1 2.5 2.0

BBB Long-Short 2.5 2.4 1.0 3.7 2.5 1.5

BB Long-Only 9.8 5.6 1.8 16.2 8.3 2.0

BB Long-Short 6.9 5.3 1.3 9.2 6.2 1.5

B Long-Only 24.9 16.1 1.5 37.2 22.0 1.7

B Long-Short 14.5 11.7 1.2 18 13.6 1.3

CCC Long-Only 52.0 45.6 1.1 65.7 58.3 1.1

CCC Long-Short 25.5 26.5 1.0 29.0 29.4 1.0

Long-Only Test Portfolio 67.1 50.4 1.3 97.0 63.2 1.5

Long-Bias 50.0 42.6 1.2 59.4 47.2 1.3

Long-Bias w/Lumps 83.1 84.2 1.0 91.0 85.5 1.1

Correlation=0.1 Correlation=0.2

(1) Single-period model uses annual PD. Multi-period model uses monthly liquidity horizon with prorated annual PD.

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1. Observations re: Results of Multi-Period Model Versus Single-Period• Multi-period model produces lower capital requirement

than single-period model– Not caused by PD, because cumulative PD over the capital

horizon is approximately the same in both approaches– No benefit of rebalancing. All exposures either default or remain

with the original credit rating. All defaults are surprises.

• Lower capital requirement is due to correlation impact– Not clear if this reduction is an authentic benefit of inter-period

diversification, or if it a problem with the multi-period model

• Capital reduction is lower as idiosyncratic risk increases– Across asset correlations– Long-short portfolios versus long-only portfolios– Across test portfolios

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1. Implication re: Default Correlation

• Link between default correlation and asset correlation in a single time period1

• Preserving the asset correlation in a multi-period simulation does not preserve the default correlation.– The effective default correlation declines significantly.– Causes a reduction in required capital

)PD(1PD)PD(1PDPDPDPDncorrelatioDefault

ondistributinormalcumulativeofInverseN

ncorrelatioassetPairwiseR),R(PD,N(PDnormal(NbivariateCumulativePD

2211

21Joint

1

21

11

Joint

))

(1) Gupton, G., Finger, C., Bhatia, M., CreditMetrics™ – Technical Document, April 1997.

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1. More Detail on Why Multi-Period Model Produces Less Capital Than Single-Period

Capital Std DevCapital ÷ Std Dev Capital Std Dev

Capital ÷ Std Dev

AAA/AA/A 1.4 0.18 8.0 N/A .17 N/ABBB 5.1 0.58 8.8 2.5 .46 5.4BB 16.2 1.95 8.3 8.3 1.32 6.3B 37.2 5.42 6.9 22.0 3.44 6.4CCC 65.7 12.12 5.4 58.3 8.93 6.5

Single-period modelwith annual PDs

Multi-period model with prorated annual PDs

• Portfolio of 87 long exposures of $1 each

• Pairwise asset correlation of 0.2 in both models

• For multi-period model: 1-month liquidity horizon and 1-year capital horizon

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1. Conclusions re: Capital Requirement for Multi-Period Versus Single-Period Model• Reasons why the multi-period model has a lower capital

requirement than a single-period model with the same PDs and asset correlation– The standard deviation is lower.

• Driven by a lower default correlation, even though the asset correlation has been preserved via the equation on page 9

• Lower default correlation is due to the nonlinear nature of the transformation from asset correlation to default correlation.

– The ratio of the 99.9% confidence interval to the standard deviation is lower.

• Driven by the assumption of zero correlation between time periods• The addition across time periods of multiple uncorrelated losses

makes the tails thinner (closer to normal). This is explained by the Central Limit Theorem.

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2. Changing the Capital Horizon (CH)

See Note 1 Corr=0.1 Corr=0.2 Corr=0.1 Corr=0.2

BBB Long-Only N/A N/A N/A N/A

BBB Long-Short N/A N/A N/A N/A

BB Long-Only 1.3 1.5 3.0 2.7

BB Long-Short 1.0 1.4 2.1 2.3

B Long-Only 1.5 1.5 3.3 2.5

B Long-Short 1.3 1.4 2.5 2.3

CCC Long-Only 1.6 1.5 3.8 3.1

CCC Long-Short 1.6 1.5 3.3 3.0

Long-Only Test Portfolio 1.4 1.5 2.6 2.4

Long-Bias 1.3 1.3 2.3 2.1

Long-Bias w/Lumps 1.2 1.2 1.7 1.7

Capital for CH = 1 qtr ÷ Capital for CH = 1 month

Capital for CH = 1 year ÷ Capital for CH = 1 month

(1) Liquidity horizon equals 1 month. PDs within the liquidity horizon based on Moody’s data

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2. Changing the Capital Horizon

• Primary driver: Longer capital horizon increases capital requirement

• Increase appears to be the net impact of two factors– If default correlation were fixed, changing the number of time

periods would change capital according the square root of the number of periods.

– Combined impact is lower than “square root” rule due to the factors identified on page 18.

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3. Changing the Liquidity Horizon with Fixed Cumulative PD over the Capital Horizon

See Note 1 Corr=0.0 Corr=0.1 Corr=0.2

BBB Long-Only 1.3 1.3 2.0

BBB Long-Short 1.0 1.2 1.5

BB Long-Only 0.9 1.8 2.0

BB Long-Short 1.0 1.3 1.5

B Long-Only 1.0 1.5 1.7

B Long-Short 1.0 1.2 1.3

CCC Long-Only 0.9 1.1 1.1

CCC Long-Short 0.9 1.0 1.0

Long-Only Test Portfolio 0.9 1.3 1.5

Long-Bias 0.9 1.2 1.3

Long-Bias w/Lumps 0.9 1.0 1.1

Capital(LH=12, CH=12) divided by capital(LH=1, CH=12)

(1) Single-period model uses annual PD. Multi-period model uses monthly liquidity horizon with prorated annual PD.

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3. Changing the Liquidity Horizon with Fixed Cumulative PD over the Capital Horizon• Across each row of the table, the cumulative PD is the

same– Monthly PD equals prorated annual PD. Therefore, cumulative PD

over 1-year capital horizon is approximately the annual PD• Therefore, the impact is entirely due to correlations

– No impact of changing liquidity horizon if correlation is 0– Increased correlation causes a greater impact

• Primary driver: Higher level of idiosyncratic risk produces a lower multiplier as the liquidity horizon is extended– Across asset correlations– Long-short portfolios versus long-only portfolios– Across test portfolios

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4. Changing PD with Fixed Liquidity Horizon

See Note 3 Corr=0.1 Corr=0.2 Note 1

BBB Long-Only N/A 1.7 2.4

BBB Long-Short N/A 2.3 2.4

BB Long-Only 1.6 1.8 1.7

BB Long-Short 1.7 1.8 1.7

B Long-Only 1.3 1.4 1.3

B Long-Short 1.3 1.3 1.3

CCC Long-Only 0.8 0.8 0.9

CCC Long-Short 0.8 0.8 0.9

Long-Only Test Portfolio 1.3 1.3 1.3Note 2

Long-Bias 1.3 1.3 1.3Note 2

Long-Bias w/Lumps 1.3 1.3 1.3Note 2

(1) PDPA is the prorated annual PD over a 1-month horizon. PDM is the monthly PD.

(2) Test portfolios have an average credit rating of approximately B, based on losses in 99.9% tail.

(3) 1-month liquidity horizon and 1-year capital horizon.

Capital with prorated annual PD divided by capital with actual monthly PD

MPA PDPD

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4. Changing PD with Fixed Liquidity Horizon• The capital horizon and liquidity horizon are fixed• The baseline PD over a 1-month horizon is the prorated annual PD

– The capital using that PD is compared to the capital using the actual PD over a 1-month liquidity horizon

• Square root rule is excellent predictor of capital ratio– Given fixed liquidity horizon and capital horizon, the correlation structure

is fixed– Given a fixed correlation structure, the s.d. of default losses is

approximately proportional to the square root of PD– Over moderate changes in PD, the ratio of the 99.9%-ile to the s.d. is

stable– The only apparent exception (BBB long) is extremely unstable due to

the impact of round-up in capital computations. This has the greatest impact for portfolios with the lowest number of default events.

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Caveats Regarding Study Results

• Results based only on small, stylized portfolios…– With simplified assumptions re: parameters

• Multi-period models are quite new– Results need to be reconciled with traditional, single-period

models

• There are other ways to apply the Gaussian copula framework in a multi-period context (e.g., David Li).

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Appendix: Portfolios Used for Evaluation• Constant rating portfolios

– Long-Only and Long-Short for BBB, BB, B, and CCC• Given portfolio size, too few defaults for results for A to AAA• Long-Only: 87 long positions of $1 each• Long-Short: Same as Long-Only plus 59 short positions of $1 each

• Test portfolios (same as December study)– Long-Only: 87 long exposures of various ratings– Long-Bias: add 59 short exposures

• Same long positions as Long-Only• For each rating grade short positions equal 2/3 of long positions

– Long-Bias with lumps• Same total exposure by rating class as Long-Bias, but one outsized

long exposure in rating classes BBB, BB, and B