Loss Given Default for European Surety Bonds Study Report of the Credit & Surety PML Working Group Fabio Sigrist, Werner A. Stahel September 12, 2011 Abstract Data on the Loss Given Default (LGD) of European surety bonds have been col- lected by several insurance companies. The data is presented and analyzed. This is done at the individual bond level as well as at the aggregate contractor level. Results are presented for maintenance, performance, and advance payment bonds. Further, the relationship between the LGD and characteristics of bonds and contractors is in- vestigated. In the case of individual bonds, a regression model that relates the LGD to bond characteristics is applied. Contents 1 Introduction 2 2 Illustration of Data 2 3 Statistical Results at Individual Bond Level 7 4 Statistical Results at Obligor Level 22 4.1 All obligors .................................... 22 4.2 Obligors with NBond>5 ............................. 30 4.3 Obligors with FV>100’000 ............................ 31 4.4 Obligors with FV>1’000’000 ........................... 33 5 Modeling Results 35 5.1 Why a quantitative model? ........................... 35 5.2 Description of the model ............................. 36 5.3 Modeling Results ................................. 38 6 Concluding Remarks 40 1
Transcript
Loss Given Default for European Surety Bonds
Study Report of the Credit & Surety PML Working Group
Fabio Sigrist, Werner A. Stahel
September 12, 2011
Abstract
Data on the Loss Given Default (LGD) of European surety bonds have been col-lected by several insurance companies. The data is presented and analyzed. This isdone at the individual bond level as well as at the aggregate contractor level. Resultsare presented for maintenance, performance, and advance payment bonds. Further,the relationship between the LGD and characteristics of bonds and contractors is in-vestigated. In the case of individual bonds, a regression model that relates the LGDto bond characteristics is applied.
A surety bond is a contractual agreement among three parties: the contractor or obligor,who performs a contractual obligation, the beneficiary or obligee, who receives the obliga-tion, and the surety, i.e., the insurance company, who covers the risk that the contractorfails to fulfill the obligation.
The goal of this study is to examine the loss of European surety bonds. Bonds ofcontractors which have defaulted, i.e., which where declared insolvent, are considered.The ultimate loss of such a bond is then called Loss Given Default (LGD).
For each bond, the maximum amount that is covered by the insurance company, aquantity called Face Value (FV), is a priori determined. This allows us to standardizethe LGD by dividing it by the Face Value. Consequently, it is sufficient to focus on thefraction LGD/FV which lies between 0 and 1.
In the following, mean values of the LGD/FV are presented. This is done for differ-ent types of Surety bonds. Confidence intervals quantifying the uncertainty about theestimates as well as standard deviations of the LGD/FV are also shown. Further, it isinvestigated how the loss depends on various bond characteristics. The analysis is alsodone for aggregate LGD/FV at the obligor level.
The rest of this report is organized as follows. In Section 2, the data is illustrated.In Sections 3 and 4, the LGDs of single bonds and of contractors are analyzed. Next, inSection 5, a quantitative model is presented and applied to the LGD of individual suretybonds.
2 Illustration of Data
There are three major types of bonds for which data was provided: maintenance, per-formance, and advance payment bonds. A maintenance bond usually guarantees againstdefective workmanship or materials for a specified period. A performance bond protectsthe obligee from financial loss should the contractor fail to perform the contract in ac-cordance with its terms and conditions (Surety Information Office [2011]). An advancepayment bond guarantees the beneficiary that he can claim back an advance he paid thecontractor, in case the contract is not carried out or not completed. In addition, the cat-egory other bonds groups a few other types of bonds such as, for instance, bid bonds anda hybrid maintenance-performance bonds.
A total of 10 different European insurance companies contributed to this collectionof data. The companies are Atradius, Euler Hermes, HCCI, Mapfre, Nationale Borg,QBE, SACE BT, Tryg Garanti, R&V, and Zurich. In addition, the following companiesparticipated in the study: Allianz, Ariel Re, Axis Capital, Hannover Re, Munich Re,Partner Re, SCOR, Swiss Re, and XL Re.
Concerning maintenance bonds, one company provides a overproportionally large num-ber of German bonds. It is believed that this overrepresentation would distort the results.Therefore, only a randomly chosen subsample of these German maintenance bonds is used.The number of bonds in the subsample is chosen so that it corresponds to the total numberof the maintenance bonds provided by all other insurance companies. Around 10% of theoriginal sample is used. We have done the analysis for different random subsamples anddid not find a dependence of the result on the specific random subsample.
The data set then contains 12621 surety bonds. We exclude bonds that have a FaceValue of less than e1000. Further, we only consider bonds whose claim files are closed. Inthe case of open bonds, the companies provide estimates of the ultimate loss. However,
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in many cases these estimates are rather crude, for instance by estimating the LGD/FVas a quarter. It does not seem reasonable to draw conclusions from such estimates. Somecompanies also provide data about bonds that were called without an insolvency of thecontractor. For the sake of consistency, these bonds were also excluded. In the end, thisleaves us with a data set consisting of 8515 surety bonds.
Percentage Number
Maintenance 79 % 6718Performance 14 % 1219
Advance Payment 5 % 400Other 2 % 178
Table 1: Type of bonds (in percent and absolute numbers).
Concerning the different types of bonds, in Table 1, the relative frequency and absolutenumbers of the types are summarized. We remark that the largest number of bonds (6718)are maintenance bonds. There are 1219 performance, 400 advance payment bonds, and178 other bonds.
Figure 1: Histogram of Face Value.
As mentioned above, the Face Value denotes the maximum amount which the issuer ofthe bond may pay to the beneficiary in case the bond is called. In Figure 1, we illustratethe Face Values of the surety bonds with a histogram. Note that a logarithmic scale isused on the x-axis. The distribution of the Face Value is considerably right-skewed (evenon the logarithmic scale) meaning that there are a lot of smaller bonds and fewer largerbonds. The mean of the Face Value is at e62225 and the median FV is e5250.
In Figure 2, the Face Values are illustrated for the different types of bonds. Withan average of e19582, maintenance bonds clearly have lower Face Values than the otherbonds. Performance bonds have an average Face Value of e154666, and for advancepayment bonds, this value is e167038.
A priori, we know that the Loss Given Default is non-negative and smaller than or
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Figure 2: Histogram of Face Value for different bond types.
equal to the Face Value0 ≤ LGD ≤ FV. (1)
In practice, one sometimes observes violations of the inequalities. This is mainly due tothe following reasons. First, there exists the possibility that an insurance company makesa negative loss, i.e., a profit. Secondly, the ultimate loss for the insurer can be greaterthan the Face Value since, e.g., the insurer might have to cover the entire amount up tothe Face Value but has additional juridical costs. In the relatively few cases where LGDis below zero or above the FV, we set it to zero or to the FV, respectively.
When knowing the Face Value, the LGD can be standardized so that it lies between 0and 1
0 ≤ LGD
FV≤ 1. (2)
In the following, we will just consider this quantity. Inference on the non-standardizedLGD is then straightforward.
In addition, for each bond we have information about the following characteristics, onwhich we comment subsequently:
• Issuance, default, and maturity date
• Size of contractor
• Experience of contractor
• Jurisdiction
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• Industry category
• Face Value in percent of underlying contract amount (FVPA)
• Conditional / first demand
• Public / private beneficiary
Usually, bonds run only over a pre-specified period of time. This means that for eachbond we know when it expires. This date is called the maturity date. Moreover, itsissuance date as well as its default date are also known. We define the Relative DefaultTime (RDT) as
RDT =default date− issuance date
maturity date− issuance data. (3)
Figure 3: Illustration of Relative Default Time (RDT).
In other words, the RDT is the time that has passed from issuance to default over thetotal lifespan of a bond. If no date of maturity is contractually determined, an estimatedvalue is provided by the insurance companies. Figure 3 illustrates the RDT. In Figure 4,the relative frequencies of the RDTs in the data is shown. The values between 0 and 1 arequite equally distributed. Then there are a few bonds which have RDTs that are above1. This can happen in cases where the maturity date is estimated. Note that we use anonlinear transformation to map the RDT between 1 and the largest observed RDT (9)to the interval [0, 1.5].
Figure 4: Histogram of observed Relative Default Time (RDT).
Next, the size of a contractor is coded as a categorical variable attaining three differentlevels:
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• Small: Turnover less than e10 million per year
• Medium: Turnover from e10 to e50 million per year
• Large: Turnover above e50 million per year
Similarly, the experience of a contractor is a categorical variable with the three levels:
• Low: Less than 5 years of experience at default date
• Mid: 5 to 15 years of experience at default date
• High: More than 15 years of experience at default date
In Tables 2 and 3, we report the number of bonds with small, medium, and largecontractors as well as low, mid, high experience. The statistics are shown for all types ofbonds. Overall, the data contains more bonds whose contractors are medium sized andmost bonds’ contractors have high experience.
Total Maintenance Performance Advance Payment Other
France 68 2 14 5 47Outside Europe 67 31 28 7 1Central Europe 56 31 6 17 2
Northern Europe 56 37 18 1 0Spain 28 0 25 3 0
Eastern Europe 21 11 3 7 0
Table 4: Number of bonds for different jurisdictions.
Jurisdiction stands for the jurisdiction which applies to a bond. Table 4 summarizesthe number of bonds for the different jurisdictions. Germany clearly provides the bulk
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of the data. This holds true for all types of bonds. Note that certain countries aregrouped into categories. This is done to have a larger number of bonds in a category andby assuming that bonds of certain regions behave similarly. Northern Europe refers toDenmark, Finland, Latvia, and Sweden. Belgium, Luxembourg, and Netherlands formthe Benelux group. UK contains England and Scotland and Central Europe consists ofAustria and Switzerland. Finally, the Czech Republic, Hungary, Poland, and Slovakia aregrouped into Eastern Europe.
Total Maintenance Performance Advance Payment Other
Building construction 4396 3960 304 62 70Infrastructure construction 1210 893 292 11 14
Table 5: Number of bonds for different industry categories.
Industry category denotes the field of industry in which the project that is covered bythe bond is realized. Table 5 shows the number of bonds for each industry category.
Usually, European surety bonds cover only a certain percentage of the underlying con-tract amount (FVPA). In contrast, performance bonds in the US usually cover 100 % ofthe underlying contract. For the bonds analyzed here, we do know this percentage. Thisvalue is typically rather low for most European surety bonds. In our data, about 95% ofall bonds have this percentage below 10%.
The obligation in first demand bonds arises on first demand, or on first demand sup-ported by a specified document, without any independent evidence of the validity of theclaim. Conditional bonds are guarantees which provide that payment will be made subjectto certain specified conditions (Myers [1998]).
Finally, it is distinguished whether the beneficiary is a private or a public company.
3 Statistical Results at Individual Bond Level
In the following, we present descriptive statistical results concerning the LGD of individualbonds and its relation to the factors presented in Section 2. We will calculated mostquantities in two versions. In the first way, each bond is weighted with its Face Value,and in the second way, each bond has the same weight irrespective of its Face Value. Inprinciple, it makes more sense to consider weighted quantities, since the loss of a portfolioconsisting of several bonds is a weighted sum over the bonds’ standardized losses wherethe weights are the Face Values. Therefore, if one does not include appropriate weights inthe statistical analysis, small bonds have too much influence. For the sake of completeness,we also report unweighted quantities.
In Figure 5, the weighted and unweighted distributions of the LGD/FV, are illustrated.In the left plot, each bonds weight is proportional to its Face Value. We also show weightedmeans, 95% confidence intervals for the means, and weighted standard deviations. Theweighted mean of the LGD/FV is about 19% and the weighted standard deviation isaround 33%. A large fraction of all bonds have zero losses and some bonds have fulllosses. In the unweighted case, the distribution places more mass near zero. This is due
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(a) Weighted (b) Unweighted
Figure 5: Histograms of LGD/FV. On the left, each bond is weighted according to its FV,and on the right is the unweighted analog. The numbers above the blue arrow and barrepresent the percentage of LGD/FVs being exactly zero or one, respectively. The dashedline represents the mean, and the shaded area around it is a 95% confidence interval.
to the fact that smaller bonds, i.e., bonds with smaller Face Values, tend to have smallerlosses. If no weighting is applied, the mean of the LGD/FV is about 5% and the standarddeviation is around 20%. Further, 90% of all bonds have zero losses.
Figure 6: Weighted histograms of LGD/FV for different bond types. See caption to Figure5 for more information on the plot.
Figure 6 illustrates weighted histograms of the LGD/FV for the different types of
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bonds. For maintenance bonds, a weighted average of 16% is observed. Performancebonds have an average LGD/FV of 25%, and advance payment have an average LGD/FVof 23%.
Figure 7: Unweighted histograms of LGD/FV for different bond types. See caption toFigure 5 for more information on the plot.
In Figure 7, the same quantities are plotted without weighting the bonds this time. Formaintenance, performance, and advance payment bonds, the average LGD/FVs decreasecompared to the weighted versions. Further, the probabilities of no loss are higher.
Next, we analyze how the LGD/FV depends on the Face Value. To this end, wefirst classify each bond into one of three categories according to whether FV ≤ 10′000,10′000 < FV ≤ 100′000, or FV > 100′000. The results are plotted in Figure 8. It seemsthat the LGD/FV is higher for bonds with higher Face Values. The picture looks verysimilar if one does not apply weighting (results not reported). This is due to the factthat in each category the bonds are of similar size and, consequently, have close to equalweights even if weighting is applied.
In order to investigate the impact of the Face Value on the loss in more detail, inFigure 9, a scatter plot of the LGD/FV versus the Face Value is shown. Note that theFace Value is on a logarithmic scale. The jittered points in the horizontal bars at zeroand one represent bonds with LGD/FV being exactly zero and one, respectively. Thecolored continuous lines are non-parametrically fitted means. The upper point of each linewas chosen so that there were at least 10 bonds with larger FVs than this point in thecorresponding category. Overall, the trend is the same as observed before: up to a certainpoint, bonds with higher Face Values have higher losses. Around e1’000’000 this trendappears to flatten. For performance bonds, there might even be a slight downwards trend
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Figure 8: Histograms of LGD/FV for different categories of Face Values. See caption toFigure 5 for more information on the plot.
for larger bonds. However, since the data gets sparser for larger FVs, the uncertainty aboutthe results increases. Note that the results are the same whether weighting is applied ornot since the weighting is done implicitly when including the Face Value as covariate inthe regression model.
Next, in Figure 10, a scatter plot of the LGD versus the Relative Default Time ispresented. For the calculation of the means, each bond is weighted with its Face Value.Overall, there seems to be a slight downward trend of the LGD with increasing time. Inparticular, this trend is observed for performance bonds. This is consistent with what oneexpects to happen in practice. Since, for instance, in construction, the closer a project isto its deadline at the point of default, the more of the project should have been completedalready and, consequently, the lower one expects the ultimate loss to be. For maintenancebonds, the LGD/FV increases up to a certain time and then starts to decrease again.Advance payment bonds also show a downward trend.
In Figures 11 and 12, histograms of the LGD grouped according to contractor size andexperience are shown. All results are based on weighted data. Concerning contractor size,larger contractors tend to have higher losses. For experience, there is no clear relationship.
Next, Figure 13 exhibits histograms which illustrate the relationships between theLGD and a bonds industry category. Overall, there is not too much variation betweenthe different categories. Infrastructure bonds tend to have slightly higher losses, whereasmachinery and road construction bonds seem to be on the other side of the spectrum.
In Figure 14, histograms are reported to illustrate the variation of the LGD betweendifferent countries. As seen in Section 2, apart from Germany, we do not have another sin-
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Figure 9: Scatter plot of LGD/FV versus Face Value (logarithmic scale). The jitteredpoints in the horizontal bars at zero and one represent bonds with LGD/FV being exactlyzero and one, respectively. The colored continuous lines are non-parametrically fittedmeans.
gle country with a large number of bonds. Since it is believed that for certain jurisdictions(Italy, France, Outside Europe, Spain, and Eastern Europe) the data is not representativeat all, these countries are grouped together in the category “Other”. In general, the resultsfor non-German bonds are very imprecise due to the low amount of data. It would be verybold to make statements concerning the variation among the different jurisdictions basedon this data.
Figure 15 illustrates the LGD versus the default time. For performance bonds, aconsiderable peak around the 2007/2008 financial crisis is observed. Apart from that,we do not observe any strong trend. To investigate the dependence of the LGD on thestate of the economy in more detail, Figure 16 shows the LGD versus first differencesof logarithmic Gross Domestic Product (GDP). Apparently, in times of macroeconomiccontraction, losses are higher than in times of economic growth.
Further, Figure 17 shows the relationship between the LGD/FV and the percentageof the underlying contract amount that is covered by the Face Value. Apparently, up toa certain level, the LGD/FV increases when this percentage is higher. Subsequently, therelationship becomes flatter, and for performance bonds the trend is slightly inverted.
Conditional and first demand bonds are compared in Figure 18. Consistent with whatone expects to happen, the plots show that conditional bonds have lower losses than firstdemand bonds.
Finally, Figure 19 compares bonds for private and public beneficiaries. For most of the
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Figure 10: Scatter plot of LGD/FV versus Relative Default Time. See caption to Figure9 for more information on the plot.
data, there is no information about the beneficiary (“NA”). From the few bonds that doprovide this information, we do not observe any difference between the two categories.
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Figure 11: Histograms of LGD/FV for different categories of contractor size. See captionto Figure 5 for more information on the plot.
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Figure 12: Histograms of LGD/FV for different categories of contractor experience. Seecaption to Figure 5 for more information on the plot.
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Figure 13: Histograms of LGD/FV for different industry categories. See caption to Figure5 for more information on the plot.
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Figure 14: Histograms of LGD/FV for different jurisdictions. See caption to Figure 5 formore information on the plot.
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Figure 15: Scatter plot of LGD/FV versus default date. See caption to Figure 9 for moreinformation on the plot. See caption to Figure 5 for more information on the plot.
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Figure 16: Scatter plot of LGD/FV versus first differences of logarithmic GDP. See captionto Figure 9 for more information on the plot.
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Figure 17: Scatter plot of LGD/FV versus the percentage of the underlying contractamount that is covered by the Face Value. See caption to Figure 9 for more informationon the plot.
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Figure 18: Histograms of LGD/FV for conditional and first demand bonds. See captionto Figure 5 for more information on the plot.
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Figure 19: Histograms of LGD/FV for private and public beneficiaries. “NA” refers tobonds for which there is no information on the beneficiary. See caption to Figure 5 formore information on the plot.
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4 Statistical Results at Obligor Level
In the following, descriptive statistical results concerning the aggregate LGD of obligorsare presented. For most covariates, aggregation from the bond level to the contractorlevel is straightforward. For instance, the FV of a contractor equals the sum of all FV ofits bonds, or the RDT of a contractor is calculated as the average RDT of all bonds forthat contractor. Since it is important to clearly distinguish the different bond types, theaggregation was done separately for each type of bond. Consequently, it is possible that, forinstance, the same contractor appears in the aggregated data set of both the maintenanceand the performance bonds. Again, we consider weighted as well as unweighted quantities.Overall, there are no major differences between the results at the obligor level and theones at the individual bond level presented in Section 3.
4.1 All obligors
(a) Weighted (b) Unweighted
Figure 20: Histograms of LGD/FV. On the left, each obligor is weighted according toits FV, and on the right is the unweighted analog. See caption to Figure 5 for moreinformation on the plot.
In Figure 20, the weighted and unweighted distributions of the LGD/FV are illustrated.In the left plot, each contractor weight is proportional to its accumulated Face Value. Weobserve a weighted mean LGD/FV of about 19% and a weighted standard deviation of28%. In the weighted case, the fraction of zero losses has decreased compared to the singlebonds. Again, in the unweighted case, the distribution places more mass near zero. Theaverage is around 6% and the standard deviation 19%. As for individual bonds, a highpercentage (84%) of zero losses is observed.
Figure 21 illustrates weighted histograms of the LGD/FV for different types of contrac-tors. In Figure 22, the same quantities are plotted again, without weighting the contractorsthis time.
For analyzing how the LGD/FV depends on the Face Value, we classify each obligorinto one of three categories according to whether FV ≤ 10′000, 10′000 < FV ≤ 100′000,or FV > 100′000. The histograms for each category are plotted in Figure 23.
In Figure 24, a scatter plot of the LGD/FV versus the Face Value is reported.Next, in Figure 25, a scatter plot of the LGD versus the Relative Default Time is
presented. For the calculation of the means, each contractor is weighted with its FaceValue.
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Figure 21: Weighted histograms of LGD/FV for different obligor types. See caption toFigure 5 for more information on the plot.
In Figures 26 and 27, histograms of the LGD grouped according to contractor size andexperience are shown. All results are based on weighted data.
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Figure 22: Unweighted histograms of LGD/FV for different obligor types. See caption toFigure 5 for more information on the plot.
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Figure 23: Histograms of LGD/FV for different categories of Face Values. See caption toFigure 5 for more information on the plot.
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Figure 24: Scatter plot of LGD/FV versus Face Value (logarithmic scale). See caption toFigure 9 for more information on the plot.
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Figure 25: Scatter plot of LGD/FV versus Relative Default Time. See caption to Figure9 for more information on the plot.
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Figure 26: Histograms of LGD/FV for different categories of contractor size. See captionto Figure 5 for more information on the plot.
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Figure 27: Histograms of LGD/FV for different categories of contractor experience. Seecaption to Figure 5 for more information on the plot.
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4.2 Obligors with NBond>5
In the following, we consider only obligors that have more than five bonds. Figure 28 showsweighted and unweighted histograms of the LGD/FV. The means are slightly higher andthe zero percentages lower compared to all obligors. Figure 29 illustrates the LGD/FV fordifferent types of bonds. There is no category “others” since there is not enough data.
(a) Weighted (b) Unweighted
Figure 28: Histograms of LGD/FV of obligors having more than 5 bonds. See caption toFigure 5 for more information on the plot.
Figure 29: Weighted histograms of LGD/FV of obligors having more than 5 bonds fordifferent bond types. See caption to Figure 5 for more information on the plot.
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4.3 Obligors with FV>100’000
Below, we show results for obligors with a cumulative Face Value that is larger thane100’000. Figure 30 shows the weighted and unweighted distributions of the LGD/FV,and Figure 31 distinguished between the different types. The results are similar to theones in the previous section.
(a) Weighted (b) Unweighted
Figure 30: Histograms of LGD/FV of obligors having a Face Value that is greater than100′000. See caption to Figure 5 for more information on the plot.
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Figure 31: Weighted histograms of LGD/FV of obligors having a Face Value that is greaterthan 100′000 for different obligor types. See caption to Figure 5 for more information onthe plot.
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4.4 Obligors with FV>1’000’000
Finally, we also consider obligors with a cumulative Face Value that is larger than e1’000’000.Figure 30 shows the weighted and unweighted distributions of the LGD/FV, and Figure31 distinguished between the different types. Again, there are no major differences to theprevious results. Overall and especially for maintenance bonds, we observe slightly higheraverage LGD/FVs.
(a) Weighted (b) Unweighted
Figure 32: Histograms of LGD/FV of obligors having a Face Value that is greater than1′000′000. See caption to Figure 5 for more information on the plot.
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Figure 33: Weighted histograms of LGD/FV of obligors having a Face Value that is greaterthan 1′000′000 for different obligor types. See caption to Figure 5 for more informationon the plot.
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5 Modeling Results
This section presents a statistical model for the LGD of individual surety bonds. Themodel determines which factors have an effect on the loss and how this effect looks like.
5.1 Why a quantitative model?
From the point of view of risk management, the goal is to predict as precisely as possiblethe potential loss of a contract. In our context, this means that one wants to make astatement about the behaviour of the future LGD of a bond whose claim file is not yetclosed.
If we know, for instance, the average LGD of all European bonds, we could simply takethis average amount as a prediction for the LGD of a new bond whose LGD we want topredict. This is, however, a rather crude estimate since it ignores a lot of information. Anextension of this simple estimate about the specific bond could be to split the collectedpast data into different subsets, e.g., according to the different jurisdictions, and calculatean average LGD value for each sub-dataset. When having a German bond, for instance,we know from past data that, on average, German bonds have a specific LGD, and thisaverage German LGD can then be taken as a prediction for the new German bond. Onthe other hand, it is likely that we have more information about the specific bond. Wemight know that the bonds contractor has high experience. So we might further split thedata and calculate the average LGD of German bonds with high experience. But we mightalso know the the size of the contractor and, consequently, would split up the data again,and so forth.
If we have many covariates for the bonds, this procedure is not feasible because, inthe end, the subsample is too small. Since we do indeed have information about a seriesof covariates (see Section 2), we advocate using a statistical model. Such a model takesinto account all available information about a bond simultaneously and allows for makinga probabilistic prediction for the LGD.
We briefly illustrate the functionality of a statistical model with an example. Forinstance, we have a bond from the Benelux countries. This bond might have a Face Valueof, say, e100000, its contractor might have middle experience and be of large size, theinsured fraction might be 5%, and it might be a performance bond. We then just plugin all this information into the model and get a prediction for the mean. Moreover, notonly the mean can be estimated, but the entire probability distribution. This means, forinstance, that statements such as “with a probability of 90 % the LGD of the bond willbe lower than 30 %” are possible.
A further advantage of such a model is that it treats the effects of the covariates onthe LGD simultaneously. We know, for instance, from Section 3 that bonds with largerFace Values tend to have higher LGDs and that bonds whose contractors are larger in sizeshow a tendency towards higher LGDs. In fact, contractors being of larger size have largercontracts and, consequently, bonds with larger Face Values as well. So, what is it thatreally influences the LGD? The contractors size or the Face Value? And if both do, whichdoes more, which does less, and to which extent does each covariate impact the LGD? Inorder to answer questions likes these, one needs to consider all covariates together. Thisis precisely what the model described in the following does.
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5.2 Description of the model
In this subsection, we briefly present the model which we will subsequently call the censoredgamma model. For more details we refer to Sigrist and Stahel [2011]. The readers notinterested in mathematical details may might skip this part.
In the following, the symbol Y will be used to denote the random variable which is tobe described by a model, i.e., Y corresponds to the LGD/FV.
As we have seen previously, the LGD/FV has a continuous distribution between 0 and1 and the probabilities that the LGD/FV is exactly 0 or 1 are non-negligible. Modellingsuch data is a non-trivial endeavor. One possible way is to introduce a so-called latentvariable Y ∗. In our case, such a variable can be regarded as a potential for loss. With thehelp of this latent variable, a parsimonious model, in terms of the number of parameters,can be constructed which provides a good fit to the data. This is done by assuming thatthe latent variable, which can attain any value, is observed only between 0 and 1 and thedistribution of the variable Y ∗ outside this interval (0, 1) determines the probabilities ofY (or the LGD/FV) being 0 or 1.
To be more specific, the latent variable Y ∗ is, conditional on some covariates X =(X1, . . . , Xp) ∈ Rp, gamma distributed with a shifted origin and a scale parameter thatdepends on X.
Denoting by gα,ϑ(y) the density of a gamma distributed variable with shape parameterα and scale parameter ϑ, the density of a shifted gamma distribution can be written as
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ϑαΓ(α)(y∗ + ξ)α−1e−(y
∗+ξ)/ϑ, y∗ > −ξ,
where ξ, ϑ, α > 0. If Gα,ϑ(y) denotes the corresponding distribution function
Gα,ϑ(y) =
∫ y
0
1
ϑαΓ(α)uα−1e−u/ϑdu,
the distribution function of a shifted gamma variable is Gα,ϑ(y∗ + ξ).The observed dependent variable Y , which corresponds to the LGD in our case, depends
on the latent variable as follows
Y = 0, if Y ∗ ≤ 0,
= Y ∗, if 0 < Y ∗ < 1,
= 1, if Y ∗ ≥ 1.
(4)
It follows that the distribution of such a censored gamma variable Y can be characterizedby
P [Y = 0] = Gα,ϑ(ξ),
P [Y ∈ (y, y + dy)] = gα,ϑ(y + ξ)dy, 0 < y < 1,
P [Y = 1] = 1−Gα,ϑ(1 + ξ).
(5)
For relating the covariates to the distributions of Y ∗ and Y , we assume that the scaleparameter ϑ is linked to the covariates through
log(ϑ) = X ′β. (6)
It can be shown that the expectation of Y is
E[Y |X] =ξGα,ϑ(ξ) + αϑ(Gα+1,ϑ(1 + ξ)−Gα+1,ϑ(ξ))
+ (1 + ξ)(1−Gα,ϑ(1 + ξ))− ξ, (7)
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and for a continuous Xj , one can calculate the marginal effect as
∂E[Y |X]
∂Xj=αϑ(Gα+1,ϑ(1 + ξ)−Gα+1,ϑ(ξ))βj . (8)
For modeling potential correlation among bonds from the same contractor, we extendthe above model by adding Gaussian random effects. More specifically, we assume thatthe data consists of i = 1, . . . , n contractors each having j = 1, . . . , ni bonds. The modelfor a bond j of contractor i is then
log(ϑij) = X ′ijβ + bi, (9)
where bi are iid N(0, σ2). The joint likelihood of the data is then
Ly(α,β, ξ, σ) =M∏i=1
∫ ni∏j=1
fα,ϑi,ξ(yij |bi)φ0,σ2(bi)dbi
where fα,ϑi,ξ(·) denotes the censored gamma density (5) and yij denoted the LGD/FV ofobservation ij. The integral does not have a closed form, but it can be approximated usingnumerical integration (Gauss-Hermite quadrature). The log-likelihood is then maximizedusing quasi Newton optimization.
Figure 34: Histogram of LGD/FV and fitted censored gamma model with no covariates.The numbers above the blue arrow and the solid bold lines represent the percentageof LGD/FVs being exactly 0 and 1, respectively. In parentheses are the correspondingnumbers as predicted by our model. The dashed red line represents the fitted model.
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5.3 Modeling Results
Before fitting a full model including all covariates, we fit two smaller models for the purposeof illustrating that the class of models presented above provide good fits to the LGD data.
In Figure 34, the fitted model including no covariates is presented. This means thatwe fit a censored gamma model but do not use any of the factors presented in Section 2.After fitting the model to the data (the fitted parameters are not reported), one gets adistribution of the LGD as specified in (5). The dashed red line in Figure 34 representsthe continuous density of the fitted model and the numbers in brackets represent thepercentage of LGD/FVs being exactly 0 or 1 as predicted by the model. This figureillustrates that the censored gamma distribution provides a good fit to the data.
We fit the full model using all available bond characteristics as covariates and includingrandom effects accounting for contractor correlation. For modelling the two ordered fac-torial variables size and experience, we use orthogonal polynomial contrasts. Concerningthe factorial variables type, jurisdiction, and industry category, we choose to use treat-ment contrasts with the base levels being “maintenance” in the case of type, “Germany”for jurisdiction, and “building construction” for industry category. For the Face Valueand the RDT we use polynomial splines with three knots at 3,4, and 5, and one knot at0.6, respectively. A few bonds do not provide information about certain covariates. Inthese cases, we impute the covariates with the mean for continuous variables and the mostfrequent value for categorical ones.
We illustrate the estimated effects of the covariates on the LGD as described in thefollowing. For a certain covariate k, we leave all other covariates constant, vary thecovariate k over all values that were observed in the data, and calculate the linear predictor.In doing so, we see how the linear predictor varies for the different values of covariate k.I.e., we obtain the “effect” of covariate k on the linear predictor. Note that we are notinterested in the absolute level itself of the linear predictor but only on the relative changethat is caused by covariate k when holding all other covariates fix.
Figure 35 illustrates these effects on the linear predictor scale for all covariates. On thex-axis of each plot is the covariate and on the y-axis the effect of the corresponding covari-ates. As mentioned before, the scale on the y-axis is only a relative scale for comparing theextent of the different effects, the absolute values have no interpretation. For continuouscovariates, the dashed lines are 95% confidence intervals, and for categorical variables, thewhiskers represent 95% confidence intervals. Confidence intervals are obtained by simu-lating from the approximate Gaussian distribution of the estimated parameters. In thecases where treatment contrasts are used, the base line is fixed (zero) and, consequently,does not have a confidence interval. Each effect is interpreted as the impact of a bondcharacteristic given all other covariates and holding all other covariates fix.
We observe the following effects on the LGD/FV of surety bonds. With increasingFV, the LGD/FV tends to be higher up to a certain point which is around 1’000’000.Subsequently there is a slight downward trend. With increasing RDT, the LGD/FV tendsto be lower. Change in GDP has no noticeable effect. Further, increasing experience resultsin a lower LGD/FV. There are only small differences between different contractor sizes.Even though the bonds of different types had marginally different LGD/FVs (see Figures6 and 7), bond type has no noticeable effect when accounting for other characteristics.This is mainly explained by the fact that in the data maintenance bonds have lower FVsthan performance bonds (see Figure 2). Consequently, when not accounting for the sizeof the bonds, the difference between the different types appears to be larger than it reallyis. Infrastructure bonds have slightly higher LGDs than building, road construction, and
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Figure 35: Illustration of effects on linear predictor. The dashed lines and whiskers repre-sent 95% confidence intervals for the estimated effects.
machinery bonds. There are some differences between the different jurisdiction, but theuncertainty in the estimates is large. Given the data set, it is not possible to observe a
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significant difference between the jurisdictions. Beneficiary has no noticeable effect. Next,the higher the percentage of a contract that is covered by the Face Value, the higher theLGD/FV. As seen in Section 3, there seems to be a slight downward trend in default time.Finally, there is only a marginal difference between conditional and first demand bonds.
6 Concluding Remarks
The LGD/FV of European surety bonds has been analyzed. We have seen that the averageLGD/FV is 19% and the weighted standard deviation is around 33%. These values havebeen presented for different types of Surety bonds. To quantify the uncertainty about esti-mates, confidence intervals were shown. Further, it was investigated how the loss dependson various bond characteristics. The analysis was also done for aggregate LGD/FV at theobligor level.
In a second stage, a statistical model for the LGD/FV was presented. It has beenshown that the model adequately describes the data. The model was used to determinethe impact of each bond characteristics on the LGD/FV, when leaving all other covariatesconstant.
References
James J. Myers. Guarantees of performance in the construction industry. In David Dadge,editor, International Bank Lending and Security. Center for International Legal Studies,Salzburg, Austria, 1998.
Fabio Sigrist and Werner Stahel. Using the censored gamma distribution for modelingfractional response variables with an application to loss given default. Manuscript,Seminar for Statistics, ETH Zurich, 2011. URL http://arxiv.org/abs/1011.1796.
Surety Information Office, 2011. URL http://www.sio.org/.
