11 17 06 modeling sovereign correlations

7/31/2019 11 17 06 Modeling Sovereign Correlations

1/21

17 JUNE 2011

MODELINGMETHODOLOGY

Authors

Amnon Levy

Nihil Patel

Libor Pospisil

Vojislav Sesum

Contact Us

[email protected]

[email protected]

Asia (Excluding Japan)+85 2 2916 [email protected]

Japan+81 3 5408 [email protected]

Modeling Sovereign Correlations

Abstract

Sovereign exposures comprise a large part of financial institutions credit portfolios, and areoften held due to their perceived low risk. While many sovereign exposures indeed exhibit low

default probabilities when considered on a stand-alone basis, a proper risk assessment mustalso account for correlations.

In this paper, we develop a sovereign correlation methodology which parameterizes theMoodys Analytics Global Correlation Model (GCorr) and uses sovereign CDS data to estimateparameters. With this methodology, we can determine correlations among sovereignexposures, as well as correlations between sovereign exposures and other asset classes within acredit portfolio. In addition, utilizing the GCorr factor structure allows us to capture theinterdependencies among sovereigns due to their intertwined economies.

Ultimately, our model allows risk managers to quantitatively assess the risk of sovereignexposures within a credit portfolio by taking into account both the stand-alone and theportfolio aspects of sovereign credit risk.


2/21

2


3/21

MODELING SOVEREIGN CORRELATIONS 3

Table of Contents

1 Introduction ........................................................................................................................................................... 5

1.1 Methodology Overview .........................................................................................................................................................................................5

2 Moodys Analytics Portfolio Modeling Framework .............................................................................................. 6

3 Methodology .......................................................................................................................................................... 9

3.1Definition of Sovereign Custom Indexes ..............................................................................................................................................................9

3.2Estimation Method for Sovereign R-squared Values ..........................................................................................................................................9

4 Data ...................................................................................................................................................................... 10

4.1 Sovereign CDS Data............................................................................................................................................................................................. 11

4.2Corporate CDS Data ............................................................................................................................................................................................ 13

5 Estimation of Sovereign R-squared Values and Results ..................................................................................... 13

5.1 R-squared Values for Sovereigns with CDS Data .............................................................................................................................................. 14

5.2Modeled Sovereign R-squared Values ................................................................................................................................................................14

6 Validation of the Sovereign Correlation Model .................................................................................................. 15

6.1 In-sample Validation: Cross-sectional Fit .......................................................................................................................................................... 15

6.2Out-of-sample Validation: Correlations with Corporates .................................... ...................................... ..................................... ................. 16

6.3Out-of-sample Validation: Estimating Sovereign R-squared Values Using Corporate CDS Spreads .................................... ........................ 16

7 Portfolio Analysis ................................................................................................................................................. 17

8 Conclusion............................................................................................................................................................ 17

Appendix A .................................................................................................................................................................. 18

Appendix B ................................................................................................................................................................. 20

References ................................................................................................................................................................... 21


4/21

4


5/21


1 IntroductionFinancial institutions carry a large number of sovereign debt exposures within their credit portfolios. In order toadequately assess the risk posed by sovereign exposures in the portfolio context, the institutions need to first determinethe exposures stand-alone risk, namely the probabilities of default. In addition, they must measure the exposures

correlation effects on the portfolios. This entails modeling correlations among changes in sovereigns credit qualities andbetween changes in the credit qualities of sovereigns and other borrowers in the portfolio. Sovereign debt instruments areoften included in a portfolio due purely to their perceived low stand-alone risk, while the level of correlation betweensovereign and other instrument types is not accounted for because quantifying such correlations is challenging.

Moodys Analytics has developed a model for sovereign correlations, which we introduce in this paper. In the model, weleverage the sovereign CDS market data to estimate spread-implied asset correlations and the Moodys Analytics GCorrmodeling framework to capture the complex interdependency structure of sovereign nations economies. Utilizing theGCorr model has another advantage as it allows us to estimate the correlations of sovereign exposures with otherinstruments within a credit portfolio.

The methodology for sovereign correlations provides risk managers with parameter estimates that match some of thesignificant empirical patterns found in sovereign spread-implied asset correlations. Specifically, we find that correlationsbetween two sovereigns tend to be higher than correlations between corporates from the corresponding countries. This

finding is consistent with the academic literature on sovereign CDS spreads.1

1.1 Methodology Overview

There is also considerable variation incorrelation levels across countries. Moreover, sovereigns within certain regions, such as Southern Europe or EasternEurope, are highly correlated. Some of these patterns are related to the economic concept of sovereign credit contagion,

wherein a sovereign credit crisis in one country triggers a similar crisis in neighboring countries.

This section provides a brief overview of the modeling methodology. We begin with liquid CDS spreads and convert thespreads to distance-to-default (DD) measures based on the Kealhofer-Vasicek structural model of credit risk.

2

Having calculated the spread-implied asset correlations (i.e., correlation among the DD changes derived from spreads),we integrate sovereigns into the Moodys Analytics GCorr model by assigning each sovereign a custom indexaweighted combination of GCorr Corporate country and industry factors, which describe the sovereigns systematic risk.Subsequently, we determine the sovereign R-squared values defined as sensitivities of the sovereigns credit qualitychanges to the returns of the custom indexes.

In thismodel, the correlation between changes in DD is equivalent to asset correlation. We use the changes in DD, rather thanthe changes in spreads, to calculate correlations between the sovereigns. The reason is that the volatility of spread changesdepends on the underlying entitys probability of default within the Kealhofer-Vasicek framework, while the volatility ofDD change is constant over time.

Within our methodology, the custom index incorporates information related not only to the sovereigns own countrybut also to other countries from the region where the sovereign is located. This model feature accounts forinterdependent economies and the concept of sovereign credit contagion.

3

The definition of sovereign custom indexes together with the spread-implied sovereign asset correlations allows us toestimate sovereign GCorr R-squared values. A general interpretation of a sovereigns R-squared is that it represents thepercentage of variation in a sovereigns credit quality explained by the variation in the sovereigns country and region

fundamentals, where fundamentals are defined as corporate firm assets in that country and region.

1See, for example, Longstaff et al. (2008), or other recent academic papers that analyze sovereign CDS spreads, such as Hilscher andNosbusch (2010), Augustin and Todengap (2010), Pan and Singleton (2008), or Remolona et al. (2008).2

For details on the estimation of DD from CDS spreads, please refer to Appendix B.3

An example of sovereign credit contagion is the 2010/2011 sovereign debt crisis in Greece, which is affecting other countries inEurope. Longstaff et al. (2008) provide another example of such contagion: [] Russian Default/LTCM crisis resulted in a majorfunding event in the hedge-fund industry that then translated into common liquidity-related contagion in sovereign credit spreads.


6/21

6

After applying the estimation methodology to the CDS data, we find that the R-squared values for different sovereignsare spread out over a wide range of values, indicating that sovereigns vary in the extent to which changes in theircorporate fundamentals translate to changes in their default risk. Therefore, using the same R-squared to calculatecorrelations leads to overestimating the correlations for some sovereign pairs, while underestimating it for others.

Finally, we validate our results by conducting three tests. In the first validation exercise, we compare the cross-sectionalfit of the modeled sovereign pair-wise asset correlations with the empirical spread-implied asset correlations. In the

second validation exercise, which can be viewed as an out-of-sample test, we use the spread-implied correlations betweensovereigns and corporates. We show that sovereigns that are highly correlated with other sovereigns tend to be highlycorrelated with corporates as well, indicating that sovereign and corporate credit is driven by a similar set of factors, asour model assumes. In the third test, we use both corporate and sovereign spread-implied asset correlations to computesovereign R-squared values as an alternative model specification. The estimation results of this alternative specificationbroadly match our original R-squared values.

The remainder of this paper is organized in the following way.

Section 2 describes the Moodys Analytics Portfolio Analytics framework. Section 3 outlines the Moodys Analytics methodology for estimating sovereign correlations. Section 4 describes the sovereign and corporate credit data used in the estimation and validation. Section 5 presents the estimation details and resulting GCorr R-squared estimates. Section 6 describes the different validation exercises conducted. Section 7 outlines our sovereign correlation parameters effect on portfolio risk measures. Section 8 concludes the paper. Appendix Aprovides information about the 89 sovereigns considered in this paper. Appendix B details the use of CDS spread term structure to back out the distance-to-default measure.

2 Moodys Analytics Portfolio Modeling FrameworkThere are two basic types of credit correlations: default correlations and credit migration correlations. Default correlationmeasures the extent to which the default of one borrower is related to that of another borrower. Credit migration

correlation measures the joint credit quality change, short of default, for two borrowers. We can infer the creditcorrelations of two borrowers by measuring their individual default probabilities and their asset correlation. The basicidea is intuitive: a borrower defaults when its asset value falls below the value of its obligations (i.e., its default point).The joint probability of two borrowers defaulting during the same time period is simply the likelihood of bothborrowers asset values falling below their respective default points during that period. We can determine this probabilityby knowing the correlation between the two borrowers asset values, and the individual likelihood of each borrowerdefaulting, as depicted in Figure 1.

Firm Y

Firm X

Both pay

kpays,jdefaults

kdefaults,jpays

Both default

Borrower j

Borrower k

Both pay

Both default

Figure 1 Joint default


7/21


With this setup, the joint distribution of the borrowers asset values can be specified by the marginal distributions and acopula. Alternatively, the asset values dynamics can be captured by a factor model. For example:

1i i i i i

r = +

(1)

Where

riis the asset return of borrower i,

iis the systematic factor,

iis the R-squared of borrower ithe proportion of risk that is captured by the systematic factor,

iis the idiosyncratic factor of borrower i.

The systematic factor i (also called custom index) represents the state of economy during a particular period, andsummarizes all the relevant systematic risk factors that affect the borrowers credit quality. The variable

irepresents the

borrower-specific risk (the idiosyncratic event or shock) that affects the borrowers credit quality or ability to repay itsdebt.

While the shock in the systematic factor i is the same for borrowers with the same custom index, the borrower-specific

shockiis unique to each borrower. In the model, the systematic factor

iis independent of the idiosyncratic factor

i.

and both have a standard normal distribution. Two borrowers correlate with one another when both are exposed tocorrelated systematic factors (with potentially varying degrees). Mathematically, the correlation between the changes incredit quality measure of any two borrowers, both within and across asset classes, is equal to:

4

1 1

1 1

( , ) ( , )

cov( , )

cov( , )( , )

1*1

i j

i j

i j

i j i i i j j j

i i i j j j

r r

i j i j

i j i j

corr r r corr

corr

+ +

+ +

=

=

= =

(2)

If the underlying borrowers are part of the same market and thus share the same systematic factor, the correlation isequal to the product of the square root of the two borrowers R-squared values:

1 1( , ) ( , )

cov( , )

1*1

i ji j i i j j

i j

i j

corr r r corr

+ + =

= =

(3)

Equation (1) serves as the basis for the Monte Carlo simulation, which is used to calculate portfolio credit risk. A factormodel such as equation (1) can be specified by two sets of parameters: the R-squared values of all borrowers and thecorrelations among systematic factors.

5

4The RiskFrontier

Monte Carlo simulation engine simulates correlated asset returns for each borrower i, in a normalized space where

each asset return is distributed with a standard normal distribution. SeeModeling Credit Portfoliosfor more details.

5SeeModeling Credit Portfoliosfor details about how the RiskFrontier Monte Carlo engine simulates correlated asset returns for

various asset classes.


8/21

8

Moodys Analytics version of equation (1), known as GCorr, is graphically represented in Figure 2.

Figure 2 Structure of the Moodys Analytics Global Correlation Model (GCorr)For a sovereign borrower, the set of systematic factors contains 49 country factors and 61 industry factors, as shown inFigure 3.

Figure 3 Decomposing a sovereign borrowers credit risk

SovereignNation

SystematicRisk

SovereignSpecific

Risk

49 Country Factors 61 Industry Factors


9/21


3 MethodologyThe asset return correlation between two borrowers in the GCorr framework is given by equation (2). In order tointegrate sovereigns into GCorr, we need to specify their custom indexes and develop a method to estimate their R-squared values.

3.1 Definition of Sovereign Custom IndexesWe construct the sovereignjs custom index by using a weighted combination of the GCorr country and industryfactors. Specifically, the custom index for a sovereignjwill be:

(4)

WhereIndustry

ir {i =1 to 61} are the GCorr industry factors andCountry

kr {k =1 to 49} are the GCorr country factors. The

weightsjSov

iw {i =1 to 61} are the sovereignjs industry weights and they are determined based on the GDP composition

of the region to which the sovereign belongs.

6 jSov

kwThe weights {k =1 to 49} are sovereignjs country weights. Theseweights are determined by GDP weighting countries from the sovereigns region, with an extra weight put on thesovereigns own country factor.

We determined the amount of overweighting the sovereigns own country factor by optimizing the fit between predictedand empirical sovereign asset correlations across our sample. We decided to construct the sovereign custom index usinginformation from the sovereigns region as well as from the sovereign itself. This decision was motivated by our aim totake into account interdependencies of economies and the possible sovereign contagion effect, wherein a debt crisis inone country is related to a debt crisis in countries in the same regional and economic group.

3.2 Estimation Method for Sovereign R-squared ValuesThe R-squared of a firm in the GCorr model is estimated using its asset returns, which are based on balance sheet

information, risk-free interest rates and equity returns. However, the same method cannot be applied to sovereigns. Toaddress this challenge, our methodology for estimating sovereign R-squared values relies on the correlations inferred fromthe sovereign CDS market.

To estimate sovereign R-squared values, we first calculate the pair-wise asset correlations among sovereigns. For thispurpose, we back out the probability of default from the term structure of sovereign CDS spreads using the Kealhofer-Vasicek model, using the same approach for calculating the spread-implied EDF as outlined in Dwyer et al. (2010).These probabilities of default can be readily converted to the distance-to-default (DD) measures as shown in Appendix B.Having calculated the weekly time series of DD for each sovereign, we calculate the correlation between DD changes foreach pair of sovereigns: ,.

We choose to calculate the correlations between sovereigns using DD changes rather than raw spreads because changes inspreads are non-stationary in the Kealhofer-Vasicek model, unlike changes in DD. More precisely, the volatility of thespread changes tends to be low when the default probability of the underlying obligor is low, and vice versa.

6The data on GDP levels and GDP composition were obtained from International Monetary Fund (www.imf.org) and CIA The

World Factbook (www.cia.gov).

61 49

1 1

j jSov SovIndustry Country

j i i k k

i k

w r w r = =

= +
http://www.imf.org/http://www.imf.org/


10/21

10

In the final step of the sovereign R-squared estimation, we estimate the following regression model, motivated byequation (2), on the set of all pairs of sovereigns:

,, ,

log log( ) log( ) 1 1 .( , )

k l k k l l

k l

CDS CDScorr DD DD

k SOV l SOV RSQ RSQ

i icorr

= + + = + +

(5)

In equation (5), k1 is an indicator variable, and k

, defined as log( )kRSQ , are the coefficients of the regression. The

estimates of k can be converted to estimates of R-squared values using the transformation

2.

kkRSQ e

=

We apply the OLS method to the regression model in equation (5) to estimate the R-squared values for 64 countrieswith liquid sovereign CDS spreads. We then use these results to model R-squared values for the remaining 25 countriesin our study. The details of that interpolation are given in Section 5.2.

Recall that a sovereigns R-squared measures the percentage of variation in sovereign credit risk changes explained by thechanges in the sovereigns fundamentals, where we define fundamentals as the assets of the corporate firms in thatcountry and region.

4 DataIn this section, we discuss data-related issues that arise when modeling sovereign correlations. Furthermore, we describethe specific CDS datasets we used to estimate and validate the Moodys Analytics sovereign correlation model.

Theoretically, there are four basic data sources to consider when estimating sovereign correlations: default rates, creditratings, sovereign bond data, and CDS data. Moodys Analytics uses time series of historical default rates to estimatecorrelation models for CRE, retail, and other asset classes with no available market data. However, given the smallnumber of sovereign defaults over past decades, this approach is not feasible. Credit ratings provide credit rating agencyviews of sovereign credit qualities. By design, ratings represent a long-term, or through-the-cycle, assessment of creditrisk. As a result, ratings do not change frequently enough to be used in a model of sovereign credit qualityco-movements.

Market sovereign bond yields and sovereign CDS spreads can be used to obtain a dynamic market view of sovereign

credit risk. This feature makes this type of data most suitable for estimating a sovereign correlation model. Despite issueswith sovereign CDS spreads, we decided to use them as our data source because the disadvantages of sovereign bondyields are larger. Namely, transforming market bond yields into credit quality measures involves several challenges, which

we can avoid by using CDS spreads. This removes the risk-free component of the yield and accounts for possibleembedded optionalities (e.g., early exercise) and other bond features. Another important issue affecting both sovereignCDS and sovereign bonds is liquidity.

To conclude, we use market CDS datasets to estimate and validate the sovereign correlation model. In Sections 4.1 and4.2, we describe the datasets and discuss how we addressed related issues, such as liquidity.


11/21


4.1 Sovereign CDS DataWe estimated the sovereign correlation model, introduced in Section 3, using sovereign CDS data. To validate themodel, we utilized corporate CDS data, described in Section 4.2. For our model, the data vendor Markit provided bothsovereign and corporate CDS spreads. Moodys Analytics employs the same datasets for estimation of CDS-implied EDFcredit measures.

7

There are two aspects of the CDS datasets worth discussing in more detail.

CDS spreads are functions of currency,doc clause, tier, and tenor. For each reference entity, we aggregate thesespreads across currencies, doc clauses, and tiers to obtain a single spread term structure at a given point in time. Rawspreads given to Markit by dealers differ by currency, but the differences are generally not systematic. Aggregatingacross currencies provides fuller coverage and is more robust. Aggregating across doc clauses also leads to fullercoverage and more model stability. Since spreads differ systematically by doc clause, we convert each spread into acommon doc clause. Another approach is to use the most liquid doc clauses. However, such doc clauses typicallydepend on the region and rating of the reference entity. Moreover, doc clause conventions can and do change overtime, and spreads of different doc clauses are on a less even footing with each other. Therefore, aggregating spreadsacross doc clauses enables us to limit any disadvantages and discrepancies.

The CDS spreads from Markit are quotes from contributing CDS dealers, and therefore do not represent actualtransactions. Despite this fact, we decided to use the Markit data because they provide a wider coverage than

transaction data. To avoid CDS spread quotes that are less likely to reflect sovereign credit risk, we applied severalliquidity filters to the data. These filters can remove stale quotes or focus on time periods with the largest changes incredit quality.

We restrict the time range of sovereign CDS spreads for the model estimation to period July 2008June 2010. Weselected the beginning of this period to remove pre-crisis spreads from the estimation. This is because we want to focusmainly on the credit risk component of the CDS spreads, which had a smaller impact on high quality sovereign spreadsbefore the financial crisis than during the crisis.

As shown in Figure 4, we see that 5-year cumulative spread-implied EDF credit measures for six select countries began toincrease dramatically from July 2008 onward.

Figure 4 The 5-year cumulative spread-implied EDF credit measures for select countries, January 2007July 2010

7For more information about this data, see Dwyer et al. (2010).

-1.00%

1.00%

3.00%

5.00%

7.00%

9.00%

11.00%

13.00%

2007 2008 2009 2010

France

Italy

Japan

Greece

Portugal

China


12/21

12

It is worth pointing out that the spreads and 5-year cumulative spread-implied EDF credit measures for most countrieshave been moving in tandem even prior to July 2008, as shown in Figure 5.

Figure 5 The 5-year cumulative spread-implied EDF credit measures for select countries: January 2007July 2010,log-scale

Overall, CDS spreads of varying quality are available for 89 sovereigns. To address the liquidity issue and create a datasetsuitable for estimation, we consider only sovereigns with at least 80 weekly reliable spreads over the period

July 2008-June 2010 (i.e., spreads that are neither missing nor stale). Spread time series of 64 sovereigns meet thiscondition.

We then convert the weekly CDS spreads for the 64 sovereigns over the period July 2008June 2010 to distance-to-default (DD) according to the methodology discussed in Appendix B. Subsequently, we determine weekly DD, or

measures of the sovereigns credit quality changes. To further reduce the effect of liquidity in the data, we apply anotherfilter. Namely, we select only 50% of weeks from the period (56 weeks), during which changes in DD for USA werelargest. The rationale for this filter is based on the fact that USA is one of the most creditworthy sovereigns (i.e., with lowEDF) and therefore larger changes in its DD are more likely to be due to systematic credit risk. As a result, we can expectthe liquidity component of DD changes to play a comparatively smaller role, and the credit risk component a bigger roleduring the selected weeks than during the remaining weeks.

The transformation and filters lead to the following final sovereign sample: weekly DD changes for 64 sovereigns and 56weeks with the strongest credit risk effect over the period July 2008June 2010. This final dataset is used for R-squaredestimation, using the method described in Section 5.1. In addition, we are able to utilize the estimation results tocalibrate R-squared values for the other 25 sovereigns for which reliable CDS spreads are not available, as described inSection 5.2. Overall, R-squared values are provided for 89 (= 64 + 25) sovereigns, representing of 99.5% of alloutstanding debt as of 2010.

8

Table 1

in Appendix Alists the 89 sovereigns considered in this paper, including the indication whether adequate CDSspreads were available.

8Source: Moodys Statistical HandbookCountry Credit, November 2010, Moodys Investors Service.

0.00%

0.01%

0.01%

0.02%

0.05%

0.10%

0.20%

0.39%

0.78%

1.56%

3.13%

6.25%

12.50%

2007 2008 2009 2010

France

Italy

Japan

Greece

Portugal

China


13/21


4.2 Corporate CDS DataWhile we primarily use sovereign CDS data to estimate sovereign R-squared values, we rely on corporate CDS data inour validation exercises. To construct our sample of corporate CDS data, we start with the time series of weekly CDSspreads of 2,618 North American and European firms. We apply several restrictions to this data. First, we exclude firmsnot covered in GCorr, which reduces the number of firms to 2,107.

9

The final corporate sample is approximately evenly divided between North American and European firms. The firms inthis sample are, on average, larger and safer than the firms in the original corporate sample. Furthermore, firms withavailable CDS spreads are much larger than median GCorr firms.

Then, we consider only the names that were the

most liquid through the sample period, and we exclude firms that have less than 80 reliable weekly CDS quotes over theperiod June 2008July 2010. These restrictions leave us with 364 firms as our final corporate sample, which we use inthe validation exercises.

5 Estimation of Sovereign R-squared Values and ResultsIn this section, we present the estimation details and the resulting sovereign R-squared values. We apply themethodology outlined in Section 3 to estimate R-squared values for the 64 sovereigns with sufficient CDS data. Section4.1 describes how the dataset was created. Section 5.1 summarizes the details of this estimation. In Section 5.2, wediscuss how we use the estimates to calibrate R-squared values using a comparables based approach for other 25sovereigns without adequate CDS spreads.

In Figure 6, we plotted histograms of the R-squared values for the 89 sovereigns and all pair-wise modeled sovereign

correlations, defined by formula ( , ) = ( ,).Let us highlight two main features of the R-squared values and modeled correlations. First, their median levels at 69%and 51%, respectively, are much higher than the corresponding values for corporates based on the GCorr 2010Corporate model. Second, sovereign R-squared values are dispersed over a wide range, roughly 40% to 90%, andmodeled sovereign correlations over a range of 20% to 90%. This highlights the peril of using just one correlation (orR-squared) for sovereignsit leads to overestimating correlations for some sovereign pairs, while underestimatingcorrelations for others.

Figure 6 Histograms: R-squared values for 89 sovereigns and for modeled sovereign correlations.

9We ensure that the corporate firms have GCorr correlation parameters so that we can calculate the custom index correlations and use

the GCorr R-squared.


14/21

14

5.1 R-squared Values for Sovereigns with CDS DataThe methodology introduced in Section 3.2 applies only to sovereigns with sufficient CDS data. Therefore, we firstselect the 64 sovereigns that passed the data and liquidity filters described in Section 4.1. For these sovereigns, wedetermine the two sets of inputs of the estimation methodology: all pair-wise custom index correlations ( ,),and all pair-wise correlations of DD changes,

(

,

). We use the GCorr 2010 Corporate factor covariance

matrix to calculate the pair-wise correlations among the sovereigns custom indexes. The correlations of DD changes arecalculated as empirical correlations of the time series of weekly DD changes over the 56 weeks during the period July2008June 2010, where the weeks were selected to reduce the impact of liquidity issues on the results. Dwyer et al.(2010) and Appendix B contain details on deriving DD measures from CDS spreads.

Having calculated ( ,)and ( ,)for each pair of the 64 sovereigns (2,016 unique pair-wisecorrelations), we are able to estimate the regression model defined in equation (5) with the OLS method. We carry outthe estimation in two stages. In stage 1, we use a sample of only the 20 safest sovereigns and estimate their parameters. Instage 2, the input data for the model is based on all 64 sovereigns, but we set the parameters for the 20 safest sovereignsequal to the values from stage 1. Subsequently, we can estimate parameters for the remaining 44 sovereigns.

Finally, we convert the estimated parameters into 64 sovereign R-squared values. The advantage of the two stageestimation is that the average levels of empirical and fitted correlations match for both the sample of the 20 safestsovereigns and the entire sample of 64 sovereigns. Moreover, if we did not use the two stage procedure and estimated allparameters in one step, the R-squared values for the safest sovereigns would be artificially low because they have lowerempirical correlations than riskier sovereigns due to the smaller importance of the credit risk component in their spreads.Such an outcome might lead to mismatch between empirical and modeled correlations of important sovereigns, such asUSA and Germany. The two stage method enables us to avoid these issues.

5.2 Modeled Sovereign R-squared ValuesThe sovereign R-squared values estimated using the regression model specified in equation (5) exhibit certain patterns.First, similar countries have similar R-squared values. An example of this is countries in the same region that havebroadly similar size and industry structure. Next, there is an association between the R-squared of a sovereign on oneside, and its creditworthiness and size on the other side. We measure creditworthiness by spread-implied EDF, and sizeby Gross Domestic Product (GDP) in USD. These observations motivate our approach for using the estimated 64sovereign R-squared values from Section 5.1 to calibrate R-squared values for 25 sovereigns without sufficient CDS data.

In particular, we estimate the following relationship based on the 64 R-squared values:

= , , + (6)As the input variables, we use the 5-year spread-implied EDF in June 2010, and GDP over the year 2010. Afterestimating model (6), we define comparables-based R-squared values for the 25 sovereigns as follows:

= ,, + (0.75),

= 1, ,25. (7)

The term (0.75) represents the empirical 75th percentile of the residuals from regression (5). Including this term in theequation results in more conservative comparables-based R-squared values.In order to reduce effects of extreme R-squared values on the modeled correlations, we replace 10% of the highest andlowest values of the 89 sovereign R-squared values (the 64 R-squared values estimated directly from data and 25R-squared values determined through a comparables based approach) with the 90th and 10th percentiles of thedistribution, respectively.

Figure 7 displays the 5 year spread-implied EDF versus the 64 sovereign R-squared values estimated directly from CDSdata and versus the 25 comparables based R-squared values, described in Section 5.2.


15/21


As shown in Figure 7, the comparables-based R-squared values are dispersed among the R-squared values estimateddirectly from the CDS data. In addition, the comparables based R-squared values do not appear to be biased in anyparticular direction.

Figure 7 The association of spread-implied EDFs vs. estimated R-squared valuesTo conclude, the output of the methodologies presented in Sections 5.1 and 5.2 is a list of R-squared values for 89sovereigns, which can be used together with the sovereign custom indexes defined in Section 3.1 to compute assetcorrelations according to formula (2).

6 Validation of the Sovereign Correlation ModelTo validate the sovereign correlation methodology introduced in this paper, we perform the following three exercises.

In-sample validation assessing the cross-sectional fit between correlations implied by the model and empiricalcorrelations.

Out-of-sample validation based on correlations between sovereigns and corporates. Out-of-sample validation comparing the R-squared values from Section 5.1with results of an alternative estimation

method which includes information from corporate CDS spreads.

6.1 In-sample Validation: Cross-sectional FitAs an in-sample validation exercise, we compare the modeled correlations to the empirical correlations across all pairs ofthe 64 sovereigns with adequate CDS spreads. For detailed information about this sample, see Section 4.1. For a given

pair of sovereigns kand l, the modeled correlation (( ,)) is defined using the R-squared valuesfrom Section 5.1 and the correlation of the sovereigns custom indexes. The empirical correlation ( ( ,)) iscalculated in the same way as in Section 5.1.

Given the regression equation (5) and the R-squared estimation method of Section 5.1, the average levels of modeledand empirical correlations are close at 52.3% and 51.9%, respectively. The slight difference in the average correlation

levels can be attributed to the fact that we adjust the extreme estimated R-squared values, as described in Section 5.2.Overall, the differences between empirical and modeled correlations are not substantially biased in one direction.

The rank correlation between empirical and modeled correlations across the pairs of 64 sovereigns is 75.1%. Thus, themodeled correlations preserve not only the average level of empirical correlations, but also the ranking of the sovereignpairs by empirical correlations.


16/21

16

6.2 Out-of-sample Validation: Correlations with CorporatesIn this exercise, our objective is to validate the main feature of our methodology using factors from the GCorr Corporatemodel to explain co-movements in sovereign credit quality.

We use the corporate CDS data described in Section 4.2 to perform the following out-of-sample analysis. For each of the64 sovereigns with adequate CDS data, we calculate the average empirical correlation across all pairs consisting of this

sovereign kand another sovereign: 1 , ,,=1 .We then convert the corporate CDS spreads to DD changes and for each sovereign k, we calculate its average empirical

correlation with corporates:1 , ,,=1 .

Finally, we determine the correlation of these averages across all 64 sovereigns k:

1 , ,,=1 ,

1 , ,,=1 (8)

Our methodology assumes that sovereign credit quality is affected by systematic factors similar to those of corporates. Ifthis assumption is valid, we expect sovereigns that are, on average, highly correlated with other sovereigns (i.e., sovereigns

with large loadings to the systemic factors) to be, on average, highly correlated with corporates as well because they sharethe systematic factors. In other words, the correlation in formula (8) should be high.

The value of the correlation in formula (8) is high at 84%. This result suggests that the corporate and sovereign assetreturns load on the same, or highly correlated, systematic factors, which is in line with our methodology assumption.

6.3 Out-of-sample Validation: Estimating Sovereign R-squared Values UsingCorporate CDS Spreads

To assess the robustness of our results, we employ an alternative methodology to estimate sovereign R-squared values.Rather than rely on the empirical correlations among changes in sovereigns distance-to-default (DD) measures inequation (5), we estimate the R-squared values using the empirical correlations between changes in sovereign andcorporate DD measures. Since correlations with corporates were not included in the original estimation, this approachcan be considered an out-of-sample exercise.

In particular, we estimate the following regression model:

k l

k l

corr DD DDk SOV l CORP

RSQ RSQ icorr

= + +

,, ,

log log( ) log( )( , )

(9)

We estimate this model on the sample consisting of the following.

The 20 safest sovereigns used in stage 1 of the method described in Section 5.1 364 corporates with adequate CDS data, described in Section 4.2This procedure yields a new set of R-squared values for the 20 safest sovereigns. The median level of our original stage 1R-squared estimates was 59.6%, while the median of the new set is 61.2%. The rank correlation between the originaland new R-squared estimates, across the 20 sovereigns, is 79%. Thus, the median level and the ranking of the new R-squared estimates and the original estimates are similar. In other words, using correlations between sovereigns andcorporates for estimation does not fundamentally change sovereign R-squared values.


17/21


7 Portfolio AnalysisUsing a sovereign correlation model may significantly impact portfolio risk measures, such as unexpected loss oreconomic capital. Portfolio risk can be assessed adequately only if the model accounts for the special features of sovereigncorrelations, such as a generally high level of sovereign correlations, a large variation in correlations across pairs ofsovereigns, and the tendency of sovereigns in the same geographical or economic group to be highly correlated. A simpleapproach assigning a single asset correlation value to all pairs of sovereigns should be considered with caution.

As shown in Figure 6, sovereign R-squared values are dispersed over a wide range, roughly 40%90%, and modeledsovereign correlations are dispersed over a range of 20%90%. Thus, using only one correlation level for sovereigns leadsto overestimating correlations for some sovereign pairs, while underestimating correlations for others. In addition, usingMoodys Analytics sovereign correlation parameters rather than a very coarse model allows risk managers to properlyaccount for sovereign risk concentration within a credit portfolio.

8 ConclusionTo determine the impact of sovereign credit exposures on a portfolio, an investor needs estimates of the correlationsamong various sovereign exposures, as well as the correlations of sovereign exposures with other asset classes in theportfolio. This paper develops and implements a methodology for estimating these correlations by utilizing the sovereignand corporate CDS market data and the Moodys Analytics GCorr factor model.

Specifically, we define spread-implied asset correlations for sovereigns as correlations among changes in distance-to-default measures derived from CDS spreads. Subsequently, we integrate sovereigns into the Moodys Analytics GCorrmodel by assigning each sovereign a custom index. This index is a weighted combination of GCorr Corporate countryand industry factors that describes the sovereigns systematic risk. The last step of the methodology is the R-squaredestimation, which ensures that the average levels of modeled correlations and empirical correlations match.

In this paper, we discuss the following empirical findings.

Correlations between two sovereigns tend to be higher than the correlations between two firms from thecorresponding countries.

There is considerable variation in correlation levels across countries, suggesting that using one fixed correlation levelfor all sovereigns is not prudent.

Sovereigns within geographical or economic groups tend be highly correlated.The correlation estimates presented in this paper cover 89 sovereign nations, accounting for over 99.5% of alloutstanding sovereign debt. The modeling framework can be used to calculate the correlations not only amongsovereigns, but also between sovereigns and other asset classes within a financial institutions credit portfolio. As a result,Moodys Analytics sovereign spread-implied EDF and sovereign correlation models allow risk managers to quantitativelyassess the risk of sovereign exposures within a credit portfolio by accounting for both the stand-alone and portfolioaspects of risk.


18/21

18

Appendix AThis appendix provides information about the 89 sovereigns considered in this paper, including whether adequate CDSspreads were available.

Table 1 List of sovereigns and territoriesSovereign/Territory CDS Data Sovereign/Territory CDS DataABU DHABI No GHANA No

ANGOLA No GREECE Yes

ARGENTINA Yes GUATEMALA Yes

AUSTRALIA Yes HONG KONG Yes

AUSTRIA Yes HUNGARY Yes

BAHRAIN Yes ICELAND No

BARBADOS No INDIA No

BELGIUM Yes INDONESIA Yes

BELIZE No IRAQ No

BRAZIL Yes IRELAND Yes

BULGARIA Yes ISRAEL Yes

CANADA Yes ITALY Yes

CHILE Yes JAMAICA Yes

CHINA Yes JAPAN Yes

COLOMBIA Yes JORDAN No

COSTA RICA Yes KAZAKHSTAN Yes

CROATIA Yes LATVIA Yes

CYPRUS No LEBANESE REPUBLIC Yes

CZECH REPUBLIC Yes LITHUANIA Yes

DENMARK Yes MALAYSIA Yes

DOMINICAN REPUBLIC No MALTA No

ECUADOR No MEXICO Yes

EGYPT Yes MOROCCO No

EL SALVADOR Yes NETHERLANDS Yes

ESTONIA Yes NEW ZEALAND Yes

FIJI ISLANDS No NIGERIA No

FINLAND Yes NORWAY Yes

FRANCE Yes OMAN No

GERMANY Yes PAKISTAN Yes

PANAMA Yes SRI LANKA No

PERU Yes SWEDEN Yes


19/21


Sovereign/Territory CDS Data Sovereign/Territory CDS DataPHILIPPINES Yes SWISS CONFEDERATION No

POLAND Yes TAIWAN No

PORTUGAL Yes THAILAND Yes

QATAR Yes TRINIDAD AND TOBAGO NoROMANIA Yes TUNISIA No

RUSSIA Yes TURKEY Yes

SAUDI ARABIA Yes UKRAINE Yes

SERBIA No UNITED ARAB EMIRATES No

SINGAPORE No UNITED KINGDOM Yes

SLOVAKIA Yes URUGUAY Yes

SLOVENIA Yes USA Yes

SOUTH AFRICA Yes VENEZUELA Yes

SOUTH KOREA Yes VIETNAM YesSPAIN Yes


20/21

20

Appendix BThis appendix details the use of CDS spread term structure to back out the distance-to-default measure. The approachtaken in this paper mirrors the approach described in Dwyer et al. (2010).

Estimating Distance-to-default from the CDS spreads

We start by using the term structure of CDS spreads to determine the term structure of risk-neutral probabilities ofdefault. We assume that the risk-neutral survival function is Weibull, which allows us to express the spreads in thefollowing form: () = (,0,1;),

where () represents the spread on the t-year CDS contract,LGD is the expected loss given default,

is the default free discount curve,and

0and

1are Weibull parameters characterizing the risk-neutral default probability term structure.

Subsequently, the t-year risk-neutral probability of default is() = 1 exp((0)).We use the term structure of CDS spreads to estimate 0 and 1 for each issuer, enabling us to compute the risk-neutralprobability of default using the formula above. In our estimation, we follow the industry convention and assume lossgiven default of 60% for corporates and 75% for sovereigns.

The Black-Scholes-Merton framework assumes the following relation between the risk-neutral and the physicalprobabilities of default: () = 1()+ ,

where is the market Sharpe ratio, and is the correlation of the issuer with the market. We use this formula to convert(

) to

(

). In the last step, we transform the physical probability to distance-to-default measures for each entity

(corporate and sovereign).


21/21

References

Acknowledgements

We would like to thank Heather Russelland Jing Zhang for their valuablefeedback.

Copyright 2011 Moody's Analytics, Inc.and/or its licensors and affiliates. All rightsreserved.

References

Augustin, Patrick, and Romeo Tedongap, 2010, Commonality in Sovereign

CDS Spreads: A consumption-based explanation. Stockholm School ofEconomics, working paper

Black, Fischer and Myron Scholes, 1973, The Pricing of Options andCorporate Liabilities.Journal of Political Economy, Vol. 81, No. 3, pp637-654.

Dwyer, Douglas, Zan Li, Shisheng Qu, Heather Russell and Jing Zhang,2010, CDS-implied EDF Credit Measures and Fair-value Spreads,Moodys Analytics White Paper.

Hilscher, Jens, and Yves Nosbusch, 2010, Determinants of Sovereign Risk:Macroeconomic Fundamentals and the Pricing of Sovereign Debt.Review of Finance, forthcoming

Levy, Amnon, 2008, An Overview of Modeling Credit Portfolios, MoodysAnalytics White Paper.

Longstaff, Francis A., Jun Pan, Lasse H. Pedersen, and Kenneth J. Singleton,2010, How Sovereign is Sovereign Credit Risk?,American Economic

Journal: Macroeconomics, forthcoming.

Merton, Robert C., 1974, On the Pricing of Corporate Debt: The RiskStructure of Interest Rates.Journal of Finance, Vol. 29, May, 449470.

Pan, Jun, and Kenneth J. Singleton, 2008, Default and Recovery Implicit inthe Term Structure of Sovereign CDS Spreads.Journal of Finance,forthcoming.

Remolona, E., M. Scatigna, and E. Wu, 2008, The dynamic pricing ofsovereign risk in emerging markets: Fundamentals and risk aversion.

Journal of Fixed Income17(4): 57-71.
http://www.moodyskmv.com/research/files/CDS_Implied_EDF_Credit_Measures_and_Fair_Value_Spreads_PUBLIC.pdfhttp://pages.stern.nyu.edu/~lpederse/papers/SovereignLPPS.pdfhttp://pages.stern.nyu.edu/~lpederse/papers/SovereignLPPS.pdfhttp://www.moodyskmv.com/research/files/CDS_Implied_EDF_Credit_Measures_and_Fair_Value_Spreads_PUBLIC.pdf

11 17 06 modeling sovereign correlations

Documents