Master Thesis Econometrics, Operations Research, and
Actuarial Science.
Estimating Incremental Risk Charge using
(Multi)Factor Copula Models
September 5, 2012
Johan Sanders s1632248
Supervisor Rijksuniversiteit Groningen: prof L.Spierdijk Supervisor ING: N.Ray, T.Stahlie
Abstract
This thesis will discuss and compare single and multi factor Gaussian and Student t copula models that can be used to estimate the incremental risk charge (IRC) of an unsecuritised credit portfolio. Furthermore, this thesis will evaluate how correlations between different issuers of credit products can be estimated. In addition, we will compare the expected shortfall of the credit portfolio at a 99.9% level with the IRC and evaluate how sensitive the IRC is to a changes in correlation.
1
Introduction
Since the financial credit crisis of 2008, financial regulators have expanded the scope of incremental default risk. This was done in light of the credit market turmoil where banks in particular experienced large losses in their trading books. A large part of these losses did not occur due to actual defaults but from credit migrations of credit products and their decrease in liquidity.
One of the new demands that was taken up in the regulatory framework for banks after the financial crisis is that banks have to take into account the incremental risk charge (IRC) when calculating their capital requirements. The IRC is defined as the estimated cost of the migration and default risk of all unsecuritised credit products in the trading book over a one year horizon at the 99.9% confidence level. Some of the most common products that impact the height of the IRC are corporate and sovereig bonds and credit default swaps (CDS). The migration risk of an issuer corresponds to the risk that the issuer of the credit product is downgraded (upgraded) to a lower (higher) credit rating while default risk corresponds to the risk that an issuer defaults.
As stated earlier, banks have to comply with an constant increasing number of reg-ulations. The specific regulatory framework that the Dutch banks must comply to when estimating the IRC is composed of the Basel II accord (2006), the Revision of the Basel II market risk framework (2011), Capital Requirements Directive (2010), Wet Financieel Toezicht from the Dutch Central Bank as detailed in Supervisory Regulation on Solvency Requirements for Market Risk (2006), EBA Guidelines on the Incremental Default and Migration Risk Charge CP49 (2011) and the Basel II fundemental review of the trading book (2012). A short summary of the require-ments given in the regulatory framework can be found in appendix A.1.
capital to cover operating losses while still being able to honor withdrawals. The capital requirement under the regulatory framework consists of the 10 day value at risk (VaR) and stressed VaR at a 99% confidence level and the 1 year IRC at a 99.9% confidence level. The fact that the IRC must be calculated on a 99.9% confidence level on a yearly basis means that the height of the IRC is equivalent to a worst case event that occurs once in a thousand years.
The regulatory framework does not specify a specific model that has to be used to calculate the IRC. This presents a challenge for banks as they need to determine a model to appropriately estimate the IRC that is in compliance with the regulatory framework. Although banks are free to use any model to capture the IRC, Laurent and Gregory (2005) find that in practice models based on factor copulas are fre-quently used to determine the VaR of a credit portfolio.
The current model developed by ING to estimate the IRC makes use of the single factor Gaussian copula model. The single factor Gaussian copula model takes into account two factors, the state of the economy, which is identical to all issuers (sys-tematic risk) and the state of the individual issuer, which is different for all issuers (idiosyncratic risk). As a result, the transition of an issuers credit rating (default or migration) is dependent on the systematic risk and the idiosyncratic risk. The single factor Gaussian copula model was first suggested by Li (2000) and has become the industry standard to model credit risk. This is due to the fact that it is intuitive, fast and easy to compute compared to reduced form models such as those proposed in Duffie and Singleton (1999).
practice, such as Risk Metrics, Credit Metrics and KMV, are based on the Gaussian dependence structure. In Mashal and Zeevi (2002) and Breymann et al (2003) differ-ent dependence structures of financial data are tested. Of all the tested dependence structures they find that the Student t dependence structure offers the most realistic dependence structure for financial data. In addition, they find that this result is amplified when one goes further in the tail. Taking these facts into account we also choose to model the IRC using the Student t factor copula, which has a Student t dependence structure (Mc Neil et al 2005) .
The single factor copula models will take into account a single systematic risk and the idiosyncratic risk. The multi factor models will incorporate two additional sys-tematic factors namely the sector and geography risk of the issuer. In addition to comparing the use of these factor copula models to model the IRC this paper will also look a at various methods to determine the correlation between issuers and look at the sensitivity of the IRC to the change in correlations
In addition to estimating the IRC we will estimate the expected shortfall (ES) of the IRC. The ES is the expected loss conditional on the IRC confidence level. In our case this means that the ES is the expected loss given the fact that the loss in the set of the 0.1% worst possible scenarios. We are interested in the ES as it gives a more reliable estimate under stressed markets or extreme price fluctuations compared to taking the VaR (Yamai and Yoshiba 2005). A second reason to take the ES instead of a confidence level is due to the tail risk under VaR. The tail risk arises due to the fact that the VaR only gives the risk up to a specific point but gives no in-dication to the size of the risks beyond the confidence level (Yamai and Yoshiba 2002)
The structure of this thesis will be as follows. Section two will contain the literature review and section three will give the problem definition of the thesis. In section four we will discuss all the input data required for the estimation of the IRC. The different methods for estimating issuer correlations will be discussed in section five. Section six will explain how the financial impact of rating migrations and defaults are determined. The mechanics of factor copula model will be given in section seven while the methods to reduce the dimensions of different factor copula models are given in section eight. The sensitivities of the IRC to issuer correlations and the explanation of the expected shortfall are discussed in section nine. The results of the IRC and ES estimations under the various factor copula models are given in section ten. The final section of this thesis will contain the conclusion.
2
Literature Review
The guidelines set in the regulatory framework for the estimation of the IRC are new and have only been finalized in the second quarter of 2012. In addition, the imple-mentation of the IRC by banks has only started recently. Due to the new nature of the IRC there is hardly any (academic) literature concerning the modeling issues that surround the IRC.
The first paper to mention factor copulas as a method to value CDO’s was the writ-ten by Li (2000). In this paper Li created the basis of what would later become the standard model to value CDO’s and similar credit products by financial institutions. The model proposed by Li was the single factor Gaussian copula model. It suggested the use of a Gaussian factor copula model to estimate defaults. The parameters in the copula model were the general state of the economy and the individual firm. The model by Li is explained further in section 7.1.1.
The model by Li has since been adjusted to become more flexible. In Andersen et al (2003) and Burtschell et al (2009) the original single factor Gaussian copula model is adjusted such that the copula function could take the form of the Normal Inverse Gaussian (NIG) or Student t copula. This change in copula function would allow for more modeling flexibility compared to the factor Gaussian copula model proposed by Li.
Another important generalization was suggested by Hull and White (2004). In this paper, the single factor copula model was extended to incorporate multiple factors. Thus allowing defaults not only to be dependent on the state of the general economy and the firm specific risk but also allowing dependence due to other factors. The model proposed by Hull and White is known as the multi factor copula model.
3
Problem Formulation
In the financial industry the single factor Gaussian copula model is widely used to model the IRC, it has been suggested that the model using a single factor Gaussian copula model may not give the most accurate estimation of the IRC (Bringo et al 2010, Martin et al 2011). The criteria on the single factor Gaussian copula model is that it does not take into account the geography of the issuer nor its sector. In addition it is argued that the use of a single factor Gaussian copula could be less suitable to model financial data as the Gaussian distribution has thin tails and zero tail dependence (Bringo et al 2010, Mashal and Zeevi 2002).
To tackle the issues stated above we will investigate what effect a single factor Stu-dent t copula model and a multi factor StuStu-dent t copula model have when estimating the IRC compared to their Gaussian counterparts. We choose the factor Student t copula model as Marshal and Zeevi (2002), Schmidt(2005) and Breymonnet et al. (2003) have show that the Student t copula offers a better fit than the Gaussian copula when modeling financial data. One of the reasons that the Student t copula has a better fit than the Gaussian copula is that the Student t copula allows one to capture extreme dependent values due to the presence of tail dependence. This phenomenon of extreme dependent values is often observed in financial time series (Mashal and Zeevi 2002).
Apart from the general state of the economy and the state of the issuer, the multi fac-tor copula models, will take into account the geography and the secfac-tor of the issuer. However, the number of factors can also be easily extended further to incorporate additional factors, such as exchange rates.
Figure 1: IRC Estimation Process
AAA rated issuers 30 BB rated issuers 107
AA rated issuers 75 B rated issuers 63
A rated issuers 194 CCC rated issuers 36
BBB rated issuers 238
European issuers 414 Government sector issuers 81
North American issuers 82 Financial sector issuers 248
Rest of the World issuers 246 Other sectors issuers 414
Total number of issuers 742 Total CS01 -0.000069 Ω
Total EAD Ω Loss at portfolio default 0.3125 Ω
EAD Europe 0.637 Ω EAD Governments 0.525 Ω
EAD North America 0.075 Ω EAD Financials 0.362 Ω
EAD Rest of the World 0.288 Ω EAD Others 0.113 Ω
Table 1: Summary of the portfolio
As one can see a large amount of data is required for the estimation of the IRC. It may be unrealistic to assume that all the data is readily available. We therefore propose a method to reduce the the amount of required data to ease the modeling of the IRC. The exact method how we do this will be explained in the section ’Model Dimension Reduction’.
4
Data Analysis
model the IRC.
4.1
CS01, Tenor, Notional, Exposure at Default and
Loss Given Default
To be able to determine the height of the financial impact of credit migrations we require the CS01, tenor, notional, exposure at default (EAD) and the loss given de-fault (LGD) of each position. The CS01 gives the position sensitivity of each issuer to a one basis point change of the credit spread. The tenor gives the maturity of the position in years.
For the notional and the EAD we need to consider two different cases. Firstly, if the credit product we are considering is a bond, the notional is the face value of the bond position. The EAD for a bond is the current market value of the bond position. The second case we need to consider is when we have a CDS position. If this is the situation, the notional is the face value of the CDS position as a negative value. The EAD of a CDS position is given as the market value of the CDS position minus the face value of the position.
The LGD is a percentage of the notional that is lost at default. The LGD need not be 1 as it can be possible to recover a portion of the notional value when an issuer de-faults. The height of the LGD is based on historical observations and expert opinions and is determined by the ING credit risk department. The LGD is generally depen-dent on the issuer, the type of product and the level of seniority one has on the claims.
4.2
Liquidity Horizon and Constant Level of Risk
The liquidity horizon is the time that is needed to get rid of a specific credit position in your portfolio when the market is stressed. This can be done either by selling or hedging the position. In this paper we will consider the liquidity horizon to be three months. The three month liquidity horizon is set as a lower limit in the regulatory framework. Using this lower limit is justified since the majority of the portfolio we are considering consists of positions in highly traded bonds and CDS, which have even proven to be liquid during the financial crisis of 2008.
The regulatory framework also requires a constant level of risk. This means that after each liquidity horizon the portfolio has to be rebalanced back to the initial state. This ensures that the risk of the initial portfolio is maintained over the entire period. Since the IRC is a measured on a yearly basis, the three month liquidity horizon implies that we need to rebalance our portfolio four times (12 months/3 months).
4.3
Issuer Credit Rating
An input that determines the hight of the IRC is the credit rating of the different issuers. The credit rating for the different issuers used in this thesis are the average ratings that the rating agencies Standard&Poor’s and Moody’s assign to the issuers. If neither one of the rating agencies have a rating for a specific issuer we will use a rating that is determined internally by ING.
Rating Definition
AAA The highest rating possible, the obligator’s capacity to meet its financial commitment is very strong.
AA The obligator differs only slightly from the highest rating. the ca-pacity to meet financial commitment of the obligator is very strong. A The obligator is somewhat suspect to adverse changes in circum-stances and economic conditions. The obligators capacity to meet its financial commitment is still strong.
BBB The obligator exhibits adequate protection parameters. Adverse economic conditions are more likely to lead to a weakened capacity of the obligator to meet its financial commitment.
BB The obligator is less vulnerable to nonpayment than other specula-tive issues. It however faces large uncertainties or major exposures to adverse conditions.
B The obligator has the capacity to currently meet its financial com-mitment on the obligation, Adverse economic conditions will likely impair the obligators capacity or willingness to meet its financial commitment.
CCC The obligator is vulnerable to nonpayment within one year and depends on favorable economic conditions for the obligator to meet its financial commitment on the obligation.
D Unlike the other ratings, the rating ’D’ is not perspective and is only used when a default has occurred, not when default is only expected.
4.4
Transition Matrix
The transition matrix used to determine the probabilities of a rating migration is a modified S&P transition matrix. The S&P transition matrix is composed using data from 1982 upto the most recent data available. Furthermore, the S&P matrix is updated on a yearly basis and covers the transitions of global coorporates and financial institutions.
We have chosen to use the S&P transition matrix as these are the rating migra-tion probabilities that are used throughout ING. The modificamigra-tions we have made include imposing a floor on the default probabilities and the removing of the ’Not Rated’ state. These measures are taken as the Basel II agreement states that each rating has a minimum default probability. The minimum default probabilities are given in Table 3. The ’Not rated’ category is removed and the percentages are spread over the other categories. This measure ensures that all issuers obtain a future credit rating.
The transition matrix of the S&P gives the probabilities of a credit migration on a yearly basis, while the liquidity horizon for the calculation for the IRC is 3 months. To model for this inconsistency we scale down the transition matrix linearly using the Markov property such that it represents a 3 month period. In Kiefer ans Larson (2004) it is shown that the Markov property holds for the credit transition matrix if the period to which it is scaled is short. To be more specific, they find that the time homogeneous Markov property can be applied to a credit transition matrix and that this model will adequately describe credit migrations up to 5 years ahead. Since we are dealing with a three month period we can assume that the Markov property holds. We thus have the following.
ˆ
T4= T (1)
can be found in Table 4 and the modified 3 month S&P transition matrix that we use to estimate the IRC is given in Table 5
Rating Minimum
De-fault Probability AAA 0.01 AA 0.03 A 0.08 BBB 0.26 BB 1.02 B 4.78 CCC 26.21
Table 3: Minimum Default Probabilities in %
AAA AA A BBB BB B CCC D NR AAA 87.91 8.08 0.54 0.05 0.08 0.03 0.05 0.00 3.24 AA 0.57 86.48 8.17 0.53 0.06 0.08 0.02 0.02 4.06 A 0.04 1.90 87.29 5.37 0.38 0.17 0.02 0.08 4.75 BBB 0.01 0.13 3.70 84.55 3.98 0.66 0.15 0.25 6.56 BB 0.02 0.04 0.17 5.22 75.75 7.30 0.76 0.95 9.79 B 0.00 0.04 0.14 0.23 5.48 73.23 4.47 4.70 11.71 CCC 0.00 0.00 0.19 0.28 0.83 13.00 43.82 27.39 14.48
AAA AA A BBB BB B CCC D AAA 97.63 2.25 0.07 0.01 0.02 0.01 0.02 0.00 AA 0.16 97.41 2.29 0.10 0.01 0.02 0.01 0.01 A 0.01 0.54 97.80 1.51 0.08 0.04 0.00 0.02 BBB 0.00 0.03 1.06 97.48 1.18 0.15 0.04 0.06 BB 0.01 0.01 0.02 1.60 95.64 2.30 0.21 0.21 B 0.00 0.01 0.04 0.03 1.78 95.37 1.69 1.08 CCC 0.00 0.00 0.07 0.10 0.18 5.48 86.19 7.97
Table 5: Modified 3 Month Transition Matrix
4.5
Credit Spreads
The credit spread is the spread between the London Interbank Offered Rate (LIBOR), and the security. The riskier the security the higher the spread will be as investors will want a higher return for the additional risk they are taking. The credit spread for each rating is given in Table 6. Note that the rating ’D’ does not have a credit spread as this is the default state.
Rating Credit Spread
AAA 14 AA 53 A 125 BBB 205 BB 441 B 781 CCC 1366
Table 6: Credit Spread in Basis Points per Credit Rating
for bonds which are valued in Euro.
5
Correlations between Issuers
To be able to estimate the IRC one has to model the possible rating migrations of all issuers in the portfolio. Recent observations of the financial markets have shown that credit migrations are correlated with each other, a result that is also shown in Wilson (1998) and many other papers. An example on how correlations effect the IRC is as follows. Consider two perfectly correlated issuers, this implies that if a default occurs, both issuers will default simultaneously. If these issuers would have a perfect nega-tive correlation the opposite would hold, namely that the issuers will never default simultaneously. The correlation between the different issuers plays an important role in the estimation process of the IRC and as we will show in section 7.1 and 7.2, the correlations between issuers determine the factor weights of the factor copula models.
The correlations between issuers can be estimated in a number of ways. Firstly, they can be estimated by looking empirically observed defaults (Lucas 1995). Secondly, by looking at equity returns (Qi et al. 2008) and finally, by means of CDS spreads (Schonbucher 2000). We will compare these different methods for estimating the cor-relation between different issuers and determine an appropriate method to estimate the issuer correlations.
Using historical default data is an objective method for measuring the correlation between different issuers. However due to the fact that defaults are a rare occur-rences, especially for highly rated issuers, the amount of historical data is sparse and is likely to to offer a poor estimation (Zhou 2001).
reflect the true correlation between issuers (Schonbucher 2000). This is caused by the fact that the equity prices are effected by non measurable factors such as spec-ulation, industry bubbles etc. Another issue with equity price data is that not all credit issuers are publicly listed and thus do not publish equity data. Examples of non listed issuers are governments.
The final method of obtaining the correlation between different issuers is with the use of CDS spreads. Changes in the credit spreads contain information on the mar-kets view on the the risk of the underlying assets. If the correlation of the credit spreads of two issuers are highly correlated we can assume that the credit quality of these two issuers are also highly correlated (Schonbucher 2000). The main disad-vantages of using credit spreads is the limited availability of data for different issuers.
The portfolio for which we wish to determine the IRC contains a large number of sovereign issuers (Table 1), of whom there are no equity price quotes available, but who do have CDS quotes. We therefore choose to make use of the method based on CDS spreads to determine the asset correlation between the different issuers. In particular, we will use the the log returns of the 5 year CDS spreads of the issuers to determine the correlation. We choose the 5 year CDS spreads over other maturities as the 5 year CDS are the most liquid (European Central Bank 2008). The log returns of the credit spreads are given as follows.
ri(n) = log
CDSi(n)
CDSi(n − 1)
Where CDSi(n − 1) is the CDS spread of issuer i at time n − 1.
The linear correlation coefficient between issuers Xi and Xj is given by
ρXi,Xj =
cov(Xi, Xj)
σXiσXj
and Zimmer 2005). For some multivariate distributions zero correlation does not im-ply independence and for some distributions the correlation is not defined. Different measures of correlations that can be considered are Kendalls Tau and Spearman rho rank correlations
Since we are considering the Gaussian and Student t factor copula models, which are members of the family of elliptical copula, the dependence structure is fully deter-mined by the linear correlation (Demara&Mc Neil 2004 and Fang&Fang 2002). Hence when we are considering correlations we refer to the linear correlation coefficient.
6
Determining Credit Migrations and the
Fi-nancial Impact
This section will, without getting into the dynamics of factor copula models just yet, explain how the credit migrations of issuers are determined. Furthermore, an explanation will be given how the financial impact of rating migrations and defaults are calculated. The final paragraph of this section will explain how the use of Monte Carlo simulations allows us to estimate the IRC of our credit portfolio.
The method for the modeling the IRC under the single and multi factor copula is as follows. Let Qi be the cumulative distribution function of the variable Xi, where Xi
represents the credit change index. This implies that the value of Xi determines the
future credit rating of the issuer. Let us define Zi(R) as the threshold value, where
the credit rating of issuer i changes rating from R + 1 to R. Furthermore, let Pi(R)
be the probability that issuer i migrates to credit rating R. The probability of Pi(R)
Pi(R) = Pi(Xi≤ Zi(R)) for R = 1 Pi(Zi(R − 1) < Xi ≤ Zi(R)) for R = 2, ..., 7 Pi(Xi> Zi(R − 1)) for R = 8
We can rewrite this in such a way that we have the following
Pi(R) = Qi(Zi(R)) for R = 1 Q(Zi(R)) − Q(Zi(R − 1)) for R = 2, ..., 7 1 − Q(Zi(R − 1)) for R = 8
It is possible to determine the future credit rating given the credit change index Xi,
the cumulative distribution function Qi and the threshold values Zi(R). The future
rating F Ri of issuer i is then given as follows.
F Ri= 1, if Qi(Xi) ≤ Qi(Zi(1)) r, if Qi(Zi(r)) > Qi(Xi) ≥ Q(Zi(r − 1)), r = 2, ..., 7 8, if Qi(Xi) ≥ 1 − Qi(Zi(7)) (3)
Let us clarify this further with the use of a simple example. Consider a ’BB’ rated bond. The probability of migrating from the current rating ’BB’ to a future rating is graphically displayed in Figure 2. For example, if the credit change index Xi for the
Figure 2: Migration Graph
If we know the future rating of issuer i, there are three possible scenarios to consider. The issuer keeps the same rating, the issuer migrates to another rating or the issuer defaults. Each of these scenarios has a different financial impact (F I) on the issuer position. The financial impact of the different scenarios are as follows.
1. No rating change has occurred for issuer i. This results that there is no financial impact, thus F Ii= 0.
and old rating of issuer i respectively. Tt is the value of the tenor t given
in years and CS01it is the height of the CS01 of issuer i under the tenor t.
There are 13 different tenors which have the values of 1 day, 1,3,6 months and 1,2,3,4,5,7,10,15 and 30 years.
3. The issuer defaults. Under this scenario we need the EAD, notional (N OT ) and the LGD of the position to determine the financial impact. Let us consider two types of credit products, bonds and CDS. If the issuer of the bond defaults we lose the EAD and gain N OT × (1 − LGD). Under a CDS contract where a default has taken place, the owner of a CDS receives the notional and pays the recovered rate of the notional (N OT × (1 − LGD)). Furthermore, the owner of the CDS loses the market value of the CDS contract. As one can see under both situations we have F Ii= EADi− N OTi× (1 − LGDi).
Summarizing, if the current rating of issuer i is O, and the new rating is given to be N , we can calculate the financial impact for issuer i as follows.
F Ii= 0 if O = N 13 X t=1 CS01it× 1 − exp((CS (N )− CS(O)) × Tt) −Tt if O 6= N, and N = 2, .., 8 EADi− N OTi× (1 − LGDi) if N = 1
The financial impact of the portfolio F Ip is the sum of the individual issuer financial
impacts. The equation for calculating the financial impact of the portfolio is given below. F Ip = I X i=1 F Ii (4)
Where I is the total number of issuers in the portfolio.
yearly financial impact of the portfolio (Y F Ip)is thus given as follows. Y F Ip = 4 X l=1 F Ip,l (5)
Where F Ip,l is the financial impact of the portfolio in the lth liquidity horizon.
To estimate the level of the IRC we will make use of Monte Carlo simulations. This approach is similar to the approach suggested in Andersen et al (2002). We will simulate a large number (1 million) of possible scenario’s of annual rating migrations for each issuer. Given these simulated ratings we can calculate the Y F Ip under each
scenario using the methods described is above. By sorting the 1 million Y F Ip from
low to high and taking the value that corresponds to the 99.9th percentile gives us the estimate for the IRC.
Summarizing, if we have the values of Xi and know the cumulative distribution
function Qi we are able to determine future migrations, the financial impact of the
portfolio and are thus able to estimate the IRC. We will determine Qi and Xi by
means of factor copula’s. Details regarding factor copulas are given in section 7.
7
Factor Copula Models
7.1
Single Factor Copula Models
The simplest version of factor copula is the single factor copula. As its name sug-gests the dependence between the variables is determined by a single factor. The first factor copula was suggested by Li(2000), which we will review more extensively in section 7.1.1. As mentioned earlier, the single factor copula model will take into account the the overall state of the economy and an issuer specific risk.
The single factor copula function is given as follows. Xi = aiM +
q 1 − a2
iZi (6)
i = 1, ..., I
Where I is the number of different issuers,−1 ≤ ai< 1, Xiis the credit change index
for issuer i, M is the systematic risk and Zi is the idiosyncratic risk. M and Zi are
independent of each other and can follow any distribution, as long as they are scaled such that they have mean zero and unit variance. As one can see in (6), the values of ai are the weights assigned to the systematic risk of of issuer i.
The correlation between Xi and Xj is given by ρXi,Xj and is given as follows.
ρXi,Xj = Cov(Xi, Xj) σXiσXj = Cov(Xi, Xj) = Cov aiM + q 1 − a2iZi, ajM + q 1 − a2jZj = aiajCov(M, M ) + ai q 1 − a2 jCov(Zi, M ) + aj q 1 − a2 iCov(M, Zj) + q 1 − a2i q 1 − a2jCov(Zi, Zj) = aiaj
We have shown here that the correlation structure between the issuers determines the weights ai that are assigned to the idiosyncratic and systematic factors.
of ai can be obtained by solving the non linear equation (7). AAT − diag(AAT) = ρA− IρA (7) Where A = a1 a2 .. . ai , ρA= 1 ρX1X2 · · · ρX1Xi ρX2X1 1 · · · ρX2Xi .. . ... . .. ... ρXiX1 ρXiX2 · · · 1 ,
AT is the transpose of the matrix A, diag(AAT) = IAAT · (AAT) where IAAT is the
identity matrix of AAT and IρA is the identity matrix of ρA.
The non linear equation (7) can be solved using the Levenberg Marquardt algorithm (LMA). A short description of the LMA is given in appendix A.8.
As we have shown earlier, an enormous amount of data is needed to be able to com-pute the IRC of a portfolio. Taking the fact into account that our portfolio consists of 742 different issuers it is unrealistic to assume that we can determine all the pairwise correlations. We therefore suggest the use proxies to reduce number of dimensions and the amount of data that is needed. Another advantage of using proxies is that it greatly reduces the computation time needed for the LMA to find a solution. The details regarding dimension reduction can be found in section 8.
The advantages of using the single factor model is that it very intuitive and that is is extremely flexible as we can choose any distribution for M and Zi as long as they
are scaled such that they have zero mean and a unit variance. For any chosen M and Zi there exists a copula model. The choice of the copula model determines the
can be used.
7.1.1 Single Factor Gaussian Copula Model
The model proposed by Li (2000), the single factor Gaussian copula model, is given by equation (7), where the systematic risk M and the idiosyncratic risks Zi follow the
Standard Normal distribution. This results that Xi will have the Gaussian Copula
structure. Details of the Gaussian copula are given in appendix A.5.
The Normal distribution is given as follows. f (x) = √ 1
2πσ2 exp
−(x−µ)2
2σ2 (8)
Where we have that µ is the mean and σ is the standard deviation. Furthermore we have that µ = 0 and σ2 = 1 under the Standard Normal distribution.
Under the single factor Gaussian copula model we have that the cumulative dis-tribution function Qi has the form of the standard normal cumulative distribution
function which is given in (9).
Qi(Xi) = 1 √ 2π Z Xi ∞ e−t22 dt (9) 7.1.2 Single Factor Student t Copula Model
Under the single factor Student t copula model the credit change index Xi follows a
Student t distribution with d degrees of freedom. This results that Xi will have the
Student t copula structure. Details on the Student t copula are given in appendix A.6. The probability distribution function of a Student t distribution is given in equation (10). Unlike the name may suggest the marginal distributions of Xi do
f (x) = Γ( d+1 2 ) √ dπΓ(d2) 1 +x 2 d −d+12 (10) Where Γ(z) is the Gamma function. The Gamma function is given in equation (13).
Under the single factor Student t copula we define the credit change index as ˆXi
which is given as follows.
ˆ Xi =
√
IGXi (11)
Where Xi is given as in (7), M and Zi follow the Standard Normal distribution (8)
and where IG is a random variable from the Inverse Gamma distribution (12) with scale and shape parameters d
2.
The Inverse Gamma distribution function is defined as f (x) = β α Γ(α)(x) −α−1 exp −β x (12) Where β is the scale parameter, α is the shape parameter and where α, β ∈ R and α, β > 0. Γ(α) is the Gamma function and is given in equation (13).
Γ(α) = Z ∞
0
tα−1exp(−t) dt (13) Since we have that Xi follows the Student t distribution we have that the cumulative
distribtion function Qi is given by the Student t cumulative distribution function,
which is given below.
Qi(x) = Γ(d+12 ) √ dπΓd2 Z x −∞ 1 (1 +td2)d+12 dt (14)
7.2
Multi Factor Copula Models
The multi factor copula model is the extension of the single factor copula model. The multi factor copula model was first mentioned in Hull and White (2004). As the name states the multi factor copula model incorporates more than one systematic factor. We can therefore use it to determine the correlation structure between issuers when we take into account the additional systematic factors region and sector. The multi factor copula model as described in Hull and White is given as follows.
Xi = ai1M1+ ai2M2+ · · · + aimMm+ Zi q 1 − (a2 i1+ a2i2+ · · · + a2im) (15) i = 1, ..., I m = 1, ..., S
Where Xi is the credit change index, I is the total number of different issuers, S is the
number of different systematic risks, a2
i1+a2i2+· · ·+a2im< 1, Mmare the independent
systematic risks. Zi is the idiosyncratic risk for issuer i and is independent of Mm
and Zj. The marginal distributions Mm and Zi can be of any distribution as long as
they are scaled such that they are distributed with mean zero and have unit variance. The correlation between Xi and Xj is given by ρXi,Xj and is determined as follows.
ρXi,Xj = Cov(Xi, Xj) σXiσXj = Cov(Xi, Xj) = Cov S X m=1 aimMm+ v u u t1 − S X m=1 a2 imZi, S X m=1 ajmMm+ v u u t1 − S X m=1 a2 jmZj = S X m=1 aim v u u t1 − S X m=1 a2 jmCov(Mm, Zj) + S X n=1 ajn v u u t1 − S X m=1 a2 imCov(Mm, Zi) + S X m=1 S X n=1 (aimajnCov(Mm, Mn)) ! + v u u t1 − S X m=1 a2 im v u u t1 − S X n=1 a2 jnCov(Zi, Zj) = S X m=1 aimajm
the issuers determine the factor weights aim for the systematic risks.
As mentioned earlier, the multi factor model we propose to use takes into ac-count the general state of the economy, three different geographical locations and three different sectors. The geographical locations we take into account are Europe, North America and ’Rest of the World’. The different sectors we will consider are, Sovereigns, Financial Institutions and ’Other Sectors’. The values of aim give an indication to what extent these different systematic risks
effect the credit change index Xi of each issuer.
To be able to draw random samples from the multi factor copula we need to find the values of aim. These values can be obtained by solving the non linear
equation AAT − diag(AAT) = ρ A− IρA (16) Where A = a11 a12 · · · a1m a21 a22 · · · a2m .. . ... . .. ... ai1 ai2 · · · aim and ρA= 1 ρX1X2 · · · ρX1Xi ρX2X1 1 · · · ρX2Xi .. . ... . .. ... ρXiX1 ρXiX2 · · · 1 .
AT is the transpose of the matrix A, diag(AAT) = IAAT · (AAT) where IAAT is the identity matrix of AAT and I
ρA is the identity matrix of ρA.
7.2.1 Multi Factor Gaussian Copula
The multi factor Gaussian copula model takes the form of (15), where the marginal distributions Zi and Mm all follow the Standard Normal distribution
(8).
When this is the case, the credit change index Xi will have the Gaussian copula
structure and hence the cumulative distribution Qi will have the form of the
Gaussian cumulative distribution function (9).
7.2.2 Multi Factor Student t Copula
The multi factor Student t copula model is given in (15). Just as in the single factor case, the credit change index follows a Student t distribution with d degrees of freedom. The credit change index is now given by ˆXi and is as
follows.
ˆ Xi =
√
IGXi (17)
Where IG a random variable from the Inverse Gamma distribution (12) with shape and scale parameter equal to d
2 and Xi is as under (15) where Mm and
Zi are independent of each other and follow the Standard Normal distribution
(8).
As was the case in the single factor Student t copula, we have that the cumu-lative distribution Qi under the multi factor Student t has the form of (14).
8
Model Dimension Reduction
why we need to simplify the factor copula model, to be able to estimate the IRC. The reasons why we need to reduce the number of dimensions of the model is due to the fact that not all issuer credit spreads are available and to ease the determination of the matrix A with the Levenberg Marquardt algorithm. To reduce the number of dimensions of the factor copula models we will make use of proxies.
The use of proxies allows us to significantly reduce the number of dimensions of the IRC model and thus making it easier (and faster) to estimate the IRC. We will discuss two different methods to reduce the dimensions of the IRC calculations. The methods proposed in this paper are similar to the method suggested in Kong and Shahabuddin (2005).
In the proposed single factor copula models we have 7 different ratings, an id-iosyncratic risk and the systematic risk, the state of the economy. In addition to this the multi factor copula also has 3 sectors and 3 different regions to take into account. To reduce the number of dimensions we will introduce the notion of a identification matrix. The size of the identification matrix is 63 × 7.
spreads we will also consider the fixed correlations that are given in the Basel II agreement, details of which can be found in appendix A.2.1.
The final step is to map all the issuers in the credit portfolio to its identical counterpart (same rating, sector and region) for which we have just determined the factor weights. Under this method we thus have that issuers with the same properties will have identical factor weights.
The use of this dimension reducing method under the single factor copula model changes the number of correlations that have to be determined from 274911 to 21 and the number of factor weights that need to be estimated from 742 to 7. Under the multi factor model the number of correlations that have to be determined is reduced from 274911 to 1953 and the number of factor weight that have to be estimated from 4893 to 441.
Under the second method (DRM2) an additional restrictive assumption is made and is only applicable for a multi factor copula model. We assume that the correlations between different regions and sectors is zero. In practice this would mean that an issuer from Europe which is active in the financial sector is only effected by movements in the general economy, Europe and in the financial sector. Let us define an identification matrix as follows.
ID = Rt ⊗Er Gr Sr (18) Where we have that Rt = i7, Er = i9, Gr = i3⊗ I3, Sr = I3⊗ i3, in is a vector
of length n containing ones and Im is a m × m identity matrix.
For each of the 63 possible combinations we take a single time series of an issuer that falls in the specific category and calculate the correlations between these 63 issuers. Having the correlations of the issuers we use the Levenberg Marquardt algorythm to solve (16) and determine the factor weights. Note that under this method we have that A ◦ ID = A. As under (DRM1) we map all the issuers to their identical counterpart. Thus, all issuers with the same properties will have the same factor weights.
The use of proxies in this way allows us to further reduce the number of factor weights that have to be estimated under the multi factor copula model to 189.
9
Expected Shortfall and Sensitivities
This section will show how sensitive the IRC is to changes in issuer correlation. Furthermore, this section will elaborate on another risk measure, the expected shortfall, for the IRC.
As we have shown, under the single and multi factor copula models, the corre-lation between the issuers determines the factor weights for the factor copula. To determine the sensitivity of the IRC to the correlations, we propose to scale the correlation with factor δ and recalculate the value of the IRC for the sin-gle and multi factor Gaussian and Student t factor copula models. The new correlations are thus given as follows.
ˆ
ρXiXj = δ · ρXiXj (19)
or is the correlation given in Basel II . From (19) we can now write the following. δ · ρXiXj = δ · S X m=1 ajmaim ! = S X m=1 √ δ ajm· √ δ aim (20) From (20), we see that to test for the sensitivity of the IRC to correlations we only need to multiply the factor weights which we have calculated in (7) and (16) with the square root of δ. We will take into account four different scenarios and choose the values for δ to be 0.8, 0.9, 1.1 and 1.2. The impact of the change in correlations will be discussed in section 10.
Although using the VaR is the common measure for risks in the financial in-dustry, the use of VaR has some shortcomings. In Yamai and Yoshiba (2002) we find that the VaR disregards tail dependence and the fat tailed properties. The reason for this is that the VaR shows tail risk, it only measures the risk at a specific quantile and disregards any risk beyond the VaR level. This can result that a portfolio with a higher portential for large losses is less risky than a portfolio with a lower potential for large losses.
therefore wish to compare the expected shortfall (ES) to the the IRC.
If the loss distribution Z is a continuous function, the ES is defined as follows. ESα(Z) = E[Z|Z ≥ V aRα(Z)]
The ES of the IRC is thus the expected value of the financial impact of the portfolio, given that we are in the set of the worst 0.1% outcomes. We calculate the ES by taking the sum of all financial impacts that are in the 0.1% worst set of scenarios and then deviding this number by the amount of scenarios that make up the worst 0.1% set.
10
Results
In this section we will evaluate the different estimation methods and sensitiv-ities of the IRC and their corresponding result. This section is split into two subsections. The first section is on the IRC and its sensitivity under the single factor copula models , while the second section is on the estimate of the IRC and its sensitivities when modeled using multi factor copula models.
10.1
Results under the single factor copula models
Degrees of Freedom
4 6 10 ∞
IRC 1.670ω 1.587ω 1.511ω ω
Expected Shortfall 2.270ω 2.100ω 1.994ω 1.178ω
Table 7: IRC and Expected shortfall under the single factor copula model with Basel II correlations
Degrees of Freedom
4 6 10 ∞
IRC 2.303ω 2.217ω 2.125ω 1.460ω
Expected Shortfall 3.173ω 3.032ω 2.840ω 1.813ω
Table 8: IRC and Expected shortfall under the single factor copula model with CDS based correlation Degrees of Freedom 4 6 10 ∞ δ = 0.8 IRC 1.591ω 1.538ω 1.424ω 0.960ω ES 2.166ω 2.045ω 1.816ω 1.095ω δ = 0.9 IRC 1.626ω 1.573ω 1.479ω 0.964ω ES 2.225ω 2.049ω 1.910ω 1.135ω δ = 1.1 IRC 1.717ω 1.671ω 1.552ω 1.025ω ES 2.339ω 2.187ω 2.127ω 1.282ω δ = 1.2 IRC 1.807ω 1.730ω 1.583ω 1.054ω ES 2.471ω 2.373ω 2.223ω 1.360ω
Degrees of Freedom 4 6 10 ∞ δ = 0.8 IRC 1.979ω 1.954ω 1.796ω 1.209ω ES 2.781ω 2.643ω 2.472ω 1.493ω δ = 0.9 IRC 2.141ω 2.091ω 1.961ω 1.293ω ES 2.833ω 2.739ω 2.592ω 1.582ω δ = 1.1 IRC 2.500ω 2.419ω 2.328ω 1.565ω ES 3.434ω 3.239ω 3.079ω 2.026ω δ = 1.2 IRC 2.706ω 2.535ω 2.465ω 1.699ω ES 3.516ω 3.369ω 3.223ω 2.210ω
Table 10: IRC and Expected shortfall under the single factor copula model with CDS based correlation
We observe from all of the tables above, that the the choice of copula func-tion has a large influence on the height of the IRC. There is an approximate 50% increase in the height of the IRC when one moves from the Gaussian to the Student t distribution with 10 degrees of freedom. The IRC between the Gaussian factor copula and the Student t factor copula can be explained by fat tailed properties of the Student t distribution and the presence of tail de-pendence under the Student t factor copula. To be more specific, the factor Gaussian copula always has zero tail dependence while the Student t always has a positive tail dependence. The definition and properties of the tail depen-dence under the Gaussian and Student t copula models are given in appendix A.7.
happen simultaneously. This in in turn increases the height of the IRC. In addi-tion, an increase in correlation increases the tail dependence under the Student t factor copula models, which further strengthens the dependence structure in the tail.
Figure 3: IRC sensitivity to correlations under single factor copula. The left figure is the single factor copula model with Basel II correlations. The right figure is the single factor copula model with CDS determined correlations
When we compare the different degrees of freedom under the Student t factor copula model we find that in all cases the IRC ans ES increase when the num-ber of degrees of freedom decreases. This can be explained by the fact that a decrease in the degrees of freedom increases the thickness of the tails of the Student t distribution and also increases the tail dependence.
weights which are determined through the use of the CDS spreads. Table 11 gives the difference between the factor weights calculated through CDS spreads and the Base II correlation. A possible explanation for the differences between the correlation is that the Basel II agreement dates from 2006, prior to the financial crisis, while calculations based on the CDS spreads are more recent and incorporate the recent financial crises. In Erb et al (1994) it is shown that during stressed economic periods the correlation between issuers is sub-stantially higher than in in periods of economic growth. A second reason that could account for the differences is the fact that we only take into account 7 issuers to determine the correlations for a specific rating and that it is purely accidental that the correlations between these issuers are high. On the other hand, the Basel II correlations have been determined through an analysis of data stets from different G10 supervisors. As we have mentioned before, an increase in correlation leads to stronger dependence structure which results in a increase of the IRC. This comparison of the single factor copula models shows us the importance of estimating the issuer correlations.
Rating Difference in Factor Weight
AAA 0.171 AA 0.211 A 0.124 BBB 0.155 BB 0.298 B 0.272 CCC 0.004
10.2
Results under the multi factor copula models
The Tables 12 to 15 will give the IRC and ES as a function of ω, which is the value of the IRC under the Gaussian single factor copula model with Basel II correlations, for the various Student t multi factor copula models with different degrees of freedom. Note that if the degrees of freedom is ∞ we obtain the Gaussian factor copula model. In Tables 12 and 14 the results of the multi factor copula models where we use DRM1 for the dimension reduction are given. Tables 13 and 15 give the IRC and ES for the multi factor copula models where DRM2 is used to reduce the dimensions factor copula models.
Degrees of Freedom
4 6 10 ∞
IRC 2.574ω 2.450ω 2.419ω 1.748ω
Expected Shortfall 3.428ω 3.266ω 3.093ω 2.205ω
Table 12: IRC and Expected shortfall under multi factor copula models with DRM1
Degrees of Freedom
4 6 10 ∞
IRC 2.651ω 2.568ω 2.448ω 1.810ω
Expected Shortfall 3.580ω 3.435ω 3.245ω 2.395ω
Degrees of Freedom 4 6 10 ∞ δ = 0.8 IRC 2.270ω 2.250ω 2.161ω 1.424ω ES 2.980ω 2.925ω 2.782ω 1.798ω δ = 0.9 IRC 2.428ω 2.321ω 2.267ω 1.567ω ES 3.239ω 3.093ω 2.981ω 1.992ω δ = 1.1 IRC * * * * ES * * * * δ = 1.2 IRC * * * * ES * * * *
Table 14: IRC and Expected shortfall under multi factor copula models with DRM1
Degrees of Freedom 4 6 10 ∞ δ = 0.8 IRC 2.214ω 2.121ω 1.964ω 1.432ω ES 3.050ω 2.881ω 2.734ω 1.800ω δ = 0.9 IRC 2.428ω 2.289ω 2.237ω 1.641ω ES 3.334ω 3.090ω 2.814ω 2.135ω δ = 1.1 IRC * * * * ES * * * * δ = 1.2 IRC * * * * ES * * * *
Table 15: IRC and Expected shortfall under multi factor copula models with DRM2
Note that in the multi factor copula models we do not show any results (*) for the IRC or ES when we test for δ = 1.1 and δ = 1.2. The reason that we do not have an estimate for the IRC and ES in these cases is due to the fact that in both of these cases the constraint of
S
X
From Tables 12 up to 15 and from Figure 4 we find that an increase of the corre-lation or a decrease in the degrees of freedom increases the level of the IRC and ES under all multi factor copula models. The reason for the increase of the IRC and ES is identical to the reasoning given under the single factor copula models.
Figure 4: IRC sensitivity to correlations under multi factor copula models. The left figure is the multi factor copula model using DRM1. The right figure is the multi factor copula model using DRM2.
As one can see the differences between the factor copula models based on the DRM1 and DRM2 are relatively small. An explanation for the differences can be that the Monte Carlo simulations used for DRM1 and DRM2 are different, resulting in a different IRC. A second explanation could be that due to the LMA, the correlation matrix calculated from the estimated factor weights of the multi factor copula models under DRM1 and DRM2 are not identical, re-sulting in a different IRC and ES.
copula models allows us to differentiate further within each category. When we move from the single to the multi factor copula model, we increase the number of categories from 7 to 63. This increase in the number of catagories allows us to give a more accurate estimation of the IRC due to the fact that less infor-mation is lost when we reduce the dimensions of the factor copula model. A disadvantage of the multi factor model is that we are constricted to the use of issuer correlations that are based on CDS spreads.
11
Conclusion
The main question we want to answer in this thesis is if the height of the IRC and ES are significantly different when they are measured using multi or single factor Gaussian and Student t factor copula models. Furthermore we wish to determine how sensitive the estimation of the IRC is to changes in issuer cor-relation. This section will summarize our findings and form a conclusion.
As one has seen in section 10 the choice of copula has a large effect on the level of the IRC. Although the Gaussian factor copula is widely used in practice by financial institutions, we find that the Student t factor copula is more suitable to estimate the IRC. This choice for the Student t factor copula model is based on empirical research such as that of Mashal and Zeevi (2002) and Breymann et al (2003) and the appealing properties of the Student t copula. Unfortunately we can not statistically test if the Student t factor copula models are indeed more suitable for the estimation of the IRC than the Gaussian factor copula model as back testing of the IRC is not possible.
some drawbacks. The most obvious is that an extra parameter, the degrees of freedom, has to be estimated. Taking into account that multiple firms are modeled using the same of degrees of freedom, the degrees of freedom of the Student t copula must be chosen such that is is suitable for all issuers. Again we base our choice of the number of degrees of freedom on empirical research that has been conducted by other authors such as Luo and Schevchenko (2009) and Mashal and Zeevi(2002). From the empirical research we find that the number of degrees of freedom used to model financial data generally vary between 4 and 10.
a great impact on the estimation of the IRC. An additional disadvantages of using CDS spreads is that CDS spreads have a limited availability since not all issuers have liquid CDS spreads.
When we compare the multi factor copula models to the single factor copula model using correlations based on CDS spreads we find that the IRC increases by approximately 20%. This difference between multi and single factor copula model can be explained by the fact that the multi factor copula models allows us to differentiate further within each category. When we move from the single to the multi factor copula model, the number of categories increases from 7 to 63. This allows us to give a more accurate estimation of the IRC due to the fact that less information is lost when we reduce the dimensions of the factor copula model. The use of a multi factor model however, constricts us to the use of correlations that are based on CDS spreads.
We consider the ES as an alternative risk measure to determine issuer migra-tions and defaults risks as the VaR method shows tail risk. The VaR disregards tail dependence and the fat tailed properties of the loss distribution. These shortcomings can result that a portfolio with a higher potential for large losses can seem less risky than a portfolio with a lower potential for large losses. When one takes the ES to determine the risks associated to issuer migrations and defaults the effect of tail risk is reduced. We find that the ES is generally 30 to 35 percent higher than the IRC. This implies that there are some extreme events in the tails that the IRC fails to capture.
of the IRC is not possible. Of all the discussed models it is our opinion that a Student t multi factor copula with 6 degrees of freedom is the most appropriate model to estimate the IRC. The choice for this model is based on the facts that the Student t distribution allows one to model for tail dependence and that empirical research has shown that the Student t copula models offers a better fit for financial data compared to its Gaussian counterpart. The choice of the multi factor copula model over the single factor copula model is due to the fact that the multi factor model allows one to differentiate between more categories which increases the flexibility of the model. We choose to use 6 degrees of freedom as this is slightly lower than the average degrees of freedom we find in other empirical research. The use of 6 degrees of freedom will therefore offer a conservative estimate of the IRC. The only disadvantage of the use of the multi factor copula model is that we are limited to the use of correlations that are based on the CDS spreads.
12
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A
Appendix
A.1
Summary of the Regulatory Framework
A summary of the most important requirements given in the regulatory frame-work for the estimation of the IRC are given below.
• Positions
The IRC model has to cover all credit positions in the trading book. Se-curitisations are out of scope.
• Risk
The IRC model must include both default and migration risk of all posi-tions.
• Confidence level and horizon
The IRC model must calculate the capital charge at a 99.9% confidence level over a one year horizon.
• Constant level of risk
The IRC model must take into account a constant level of risk. This means that the portfolio must be rebalanced back to its original state at the end of each liquidity horizon and must have a constant level of risk over the one year horizon.
• Liquidity horizon
The IRC model must take into account the liquidity horizon applicable to each trading position, where the liquidity horizon has a floor of three months.
• Correlations
• Weekly calculations
The IRC should be estimated at least on a weekly basis.
A.2
Correlations
A.2.1 Correlations based on Basel II
In the Basel II accord, a standardized measure to determine the correlation between issuers with an identical rating is given. This measure is only depen-dent on the probability of default of each issuer and thus issuers with the same rating will have the same correlation with each other. The formula in Basel II for the calculating correlations is as follows.
ρ = 0.12 1 − e −50×P D 1 − e−50 + 0.24 1 −1 − e −50×P D 1 − e−50
Where the P D is the probability of default given in the S&P transition matrix on a yearly basis. The correlations calculated by the according to the method given in the Basel II agreement are given in table (16). We determine the corresponding factor weights aim as follows.
ρ = aimajm
√
ρ = aim= ajm
Rating Correlation Factor Weights AAA 0.239 0.489 AA 0.238 0.488 A 0.235 0.485 BBB 0.225 0.474 BB 0.192 0.438 B 0.131 0.361 CCC 0.120 0.346
Table 16: Basel II correlations and factor weights
A.2.2 Correlations based on issuer CDS
Under the single factor models we require 7 issuers. The correlations we use will be based on the correlations between the issuers given in Table 17. These specific issuers have been chosen as they have the greatest weight in the portfolio in their specific category.
Rating Issuer AAA Germany AA China A South Korea BBB Italy BB Hungary B Ukraine CCC Greece
Table 17: Issuers used for model dimension reduction
their specific category.
Note that we do not have an issuer for each combination of rating, region and sector. If it is the case that we have no issuer we map all the issuers of that specific category to a counterpart with the same rating and region. The choice of the counterpart is given in Table 18.
Missing Value First Choice Second Choice
Government Financial Other
Financial Government Other
Other Financial Government
Rating European Issuers
Governments Financial Other
AAA Germany B.W. Landesbank
-AA Belgium Nordea Royal Dutch Shell
A Poland Barclays Bayer AG
BBB Italy NIBC Porsche
BB Hungary Espirito Santo ThysenKrupp
B Ukraine Comercial Portugues Hellenic Telecom
CCC Greece Nat Bank of Greece
-Rating North American Issuers
Governments Financial Other
AAA Canada - Johnson&Johnson
AA United States Swiss Re General Electric
A - Morgan Stanley Caterpillar
BBB Mexico Bank of America Thomson Reuters
BB - - Toll Brothers
B - MBIA corp TRW automotive
CCC - Radian Group
-Rating Rest of the World Issuers
Governments Financial Other
AAA Australia - Temasek
AA China Westpac BKG MTR
A South Korea Bank of China Hutchison Whampoa
BBB Brazil ICICI Bank Noble
BB Philippines Arab Banking Corp
-B Argentina -
-CCC - - Shire Pharmaceuticals
A.3
Copula Functions
Let us have that C is a mapping of the form C : [0, 1]n → [0, 1], thus C is a
mapping form the unit hypercube into the unit interval. We then have that the function C(u) = C(u1, ..., ud) is a copula if the following three properties hold.
• C(ui, ..., ud) is increasing in each component ui
• C(1, ..., 1, ui, 1, ..., 1) = ui for all i ∈ 1, ..., n, ui ∈ [0, 1].
• ∀(a1, ...., an), (b1, ..., bn) ∈ [0, 1]n with ai ≤ bi we have the following. 2 X i1=1 · · · 2 X in=1 = 1(−1)i1+···+idC(u 1i1, ..., unin) ≥ 0 Where uj1 = aj and uj2 = bj ∀ j ∈ 1, ..., n.
A.4
Sklars Theorem
Let F be a joint distribution function with margins F1, ..., Fn. There then exists
a copula C : [0, 1]n→ [0, 1] such that ∀x
1, ..., xn ∈ [−∞, ∞] we have
F (x1, ..., xn) = C(F1(x1), ..., Fn(xn)) (21)
= C(u1, ..., un)
If the margins are continuous then C is uniquely defined, otherwise C is uni-wuely determined on Ran F1× · · · × Ran Fn. Where Ran Fi denotes the range
of Fi.
Conversely, if we have that C is a copula, and Fi, ..., Fnare univariate
distri-butions, then the function F as defined in (21) is a joint distribution function with margins F1, ..., Fn
A.5
Gaussian Copula
If X ∼ Nn(0, Σ) is a Gaussian random vector with correlation matrix Σ, then
its copula is the Gaussian copula. The Gaussian copula is given by CΣGa(u) = P (Φ(X1) ≤ u1, ..., Φ(Xn) ≤ un)
= ΦΣ(Φ−1(u1), ..., Φ−1(un))
Where Φ is the standard univariate normal distribution function and ΦΣis the
joint distribution function of X
A.6
Student t Copula
If X ∼ tn(v, 0, Σ) is a Student t random vector then its copula is the Student
t copula. The Student t copula is given by
Cv,Σt (u) = P (tv(X1) ≤ u1, ..., tv(Xn) ≤ un)
= tv,Σ(t−1v (u1), ..., t−1v (un))
Where tv is the standard univariate Student t distribution function with v
de-grees of freedom and tv,Σ is the joint distribution function of X and Σ is the
correlation matrix of X
A.7
Tail Dependence
Tail dependence is a measure of the strength of the dependence of two random variables X1 and X2 in the tails of their distributions. Upper tail dependence
is defined as the probability that X2 exceeds quantile q given that X1 exceeds
q. Naturally X1 and X2 are interchangeable. The following definitions with
respect to tail dependency are given in Mc Neil et al (2005).
Let us have that X1 and X2 are random variables with distribution functions
X2 is given by λu and λl respectively is as follows. λu = lim q→1−P (X2 > F −1 2 (q)|X1 > F1−1(q)) λl = lim q→0+P (X2 ≤ F −1 2 (q)|X1 ≤ F1−1(q))
If we have that the distribution functions F1 and F2 are continuous and thus
there exists a unique copula C we can write the tail dependence as follows. λl = lim q→0+ C(q, q) q (22) λu = lim q→0+ ˆ C(q, q) q (23)
Where ˆC is the survival copula of C. The relationship between a copula and its survival copula is given as follows.
ˆ
C(1 − u1, 1 − u2) = 1 − u1− u2+ C(u1, u2)
If the copula C is symmetric we must have that λu = λl since under those
copulas we have that ˆC = C.
A.7.1 Tail Dependence Gaussian Copula The Gaussian Copula CGa
ρ is a symmetric copula. Let (X1, X2) = (Φ−1(U1), Φ−1(U2))
such that (X1, X2) have a bivariate normal distribution with correlation
coeffi-cient ρ. The tail dependence is then given given as follows. λ = 2 lim x→−∞P (X≤x|X1 = x) = 2 lim x→−∞Φ x√1 − ρ √ 1 + ρ = 0
A.7.2 Tail Dependence Student t Copula Just as the Gaussian copula, the Student t copula,Ct
v,ρ , is a symmetric copula.
Let (X1, X2) = (t−1v (U1), t−1v (U2)), where tv is the univariate t distribution with
v degrees of freedom and let ρ be the correlation coefficient. The tail dependence of the Student t copula is then given as follows.
λ = 2 tv+1 −
s
(v + 1)(1 − ρ) 1 + ρ
!
From this result we can conclude that the Student t copula always has tail dependence which increases non-linearly. In Figure 5 the tail dependence for the Student t copula, with 4,6 and 10 degrees of freedom is given.
Figure 5: Studnet t copula tail dependence
A.8
Lenvenberg-Marquardt Algorithm
variables (xi, yi), the parameters α of the model curve f (x, α) such that the
sum of the squares of the deviations S(α) is minimal. Where we have that S(α) is as follows. S(α) = m X i=1 (yi− f (xi, α))2
A.8.1 The Solution of the Levenberg Martquardt
The LMA is a an iterative procedure. To initiate the minimization one has to provide an initial guess for the vector α. In each iteration the vector α is replaced by a new estimate namely α + δ. To determine δ, the function f (xi, α + δ) is approximated by their linearizations.
At the minimum of S(α) the gradient with respect of S to δ will be zero. The approximation of f (xi, α + δ) given in vector notation is as follows.
S(α + δ) ≈ ||y − f (α) − J α||2
Where J is the Jacobian matrix and where f, y are the vectors corresponding to f (xi, α) and yi respectively. When we take the derivative with respect to δ
and setting the outcome equal to zero we find the following.
(JTJ )δ = JT[y − f (α)] (24)
Levenberg and Marquardt introduced the use of a damping parameter fac-tor λ diag(JTJ ) changing equation (24) into the following equation, which is
known as the LMA.
(JTJ ) + λ diag(JTJ )δ = JT[y − f (α)]
