SOVEREIGN RATINGS IMPLIED BY COUPLED CDS-BOND MARKET DATA
STEFANO MARMI, ALDO NASSIGH, AND DANIELE REGOLI
Abstract. We propose an approach to sovereign market implied ratings based on informations coming
both from Credit Default Swap spreads and bond spreads in a unified way. Operationally speaking, we
implement a Support Vector Machine type of selection in the plane CDS-bond. Our numerical results
seem to confirm that introducing the bond dimension accounts for implied ratings more accurate and
with greater predictive power with respect to the 1-dimensional CDS implied ratings.
Contents
1. Introduction
2. Market implied ratings: various existing approaches
3. sovereign market implied ratings: a CDS penalty-function analysis
4. CDS+bond based analysis
5. statistical analysis: SVM vs. KV implied ratings
5.1. accuracy
5.2. precision & recall
5.3. gap-conditioned analysis
6. Conclusions
Appendix A. data details
Credit Default Swaps
Bonds
Agency ratings
References
1
2
3
5
10
11
15
17
19
24
24
24
25
25
1. Introduction
Ratings provided by credit rating agencies (CRAs) are meant as indicators/opinions about the issuer’s
ability to pay back their debt in due time [1, 2, 3]. Their role in the modern financial system could be
hardly overestimated. They are of key importance in at least three respects: for investors, who need
information on the risks in which they are going to incur if investing in certain products; for issuers of
financial products, who need ratings in order to attract investors and inform them on their creditworthiness; finally, credit ratings have become increasingly important in regulatory policies: indeed, financial
institutions can use credit ratings from approved agencies in order to assess their capital requirements.
Given that their importance is out of doubt, nevertheless there is plenty of criticism about CRAs and
ratings themselves, in particular after their crucial role during the 2008 mortgage crisis and the 2011
euro-zone crisis.
Leaving aside the structural critics, one of the main concerns about agency ratings is their (physiological) delay in capturing the actual health or creditworthiness of the issuer. This is the reason why, aside
and complementing the information of agency ratings, the idea of capturing some kind of early warnings
Date: November 26, 2014.
JEL Classification. G24, C44, C55.
Key words and phrases. credit ratings, market implied ratings, support vector machine, credit default swaps, default
risk.
1
2
STEFANO MARMI, ALDO NASSIGH, AND DANIELE REGOLI
extracted by the market has become increasingly popular. These warnings are known as market implied
ratings or “point in time” ratings.
It is worth to stress that agency ratings and market implied ratings should be considered as complementary to each other: the stability in time of agency ratings and their independence of short term
market movements are important features, that investors and emitters look for. At the same time, in
particular for the investor side and for portfolio management, the market is a source of high frequency
information that should not be neglected and can very well suit the role of giving sort of temperature to
have a prompt idea of the ’health’ of debt emitters.
It is in this spirit that market implied ratings have grown in importance and popularity, and that are
now concocted and sold by CRAs themselves besides the ’standard’ ratings (see, e.g. [4, 5, 6, 7, 8, 9, 10]).
In this paper we focus on market implied ratings for sovereign debt emitters and we propose the idea of
using both the market of Credit Default Swaps and that of bonds. Indeed, in [11] qualitative arguments are
presented, by which, for sovereign debt emitters and in particular in situations in which various countries
share a common currency as in the Euro area, CDS alone may not capture the exact information about
the real creditworthiness of a state. As far as we know, ratings derived by market data have up to now
been calculated with one single source of information as input, namely ratings implied by CDS or bonds
or equity. The original proposal of this work is precisely to present a way of calculating market implied
ratings taking into account in one single procedure two different sources of market information. The
procedure proposed is nonetheless sufficiently flexible to be generalized to different sectors and different
input variables.
The paper is structured as follows: section 2 briefly reviews the literature on market implied ratings;
in section 3 we derive CDS-implied ratings for our panel of sovereign countries, using the technique of [4]
and [5]; we describe our proposal in section 4, and then calculate the implied ratings for our dataset;
finally, section 5 is devoted to the statistical analysis of the performances of the two sets of implied
ratings. Concluding remarks follow.
2. Market implied ratings: various existing approaches
Most of the literature on market implied ratings refers to corporate agents and typically deals with
Credit Default Swap spreads, bonds and equity data. The first attempt in this direction dates back to
1974, with the so-called Merton model [12] whose idea is to see equity as a call option on assets with debt
as strike, use equity and equity volatility market data to invert the relation, thus finding the asset value
and asset volatility of the firm: from assets and debts a “distance to default” is calculated, which then
can be mapped to a probability of default and to a rating score. This methodology, which is at the core
of what are now called the “structural” models for default probabilities, has become very popular during
the years, and it is actually implemented by both Moody’s and Fitch as an early warning tool over their
long-term ratings (Moody’s KMV and Fitch EIR [9, 8]).
Other, non-structural, attempts make a direct use of CDS or bond spreads, which are indeed a proxy
of the credit risk of the issuer (see, for instance [13]). One of the first proposals (2003) can be found in
[4] and is based on the assumption that “agency ratings are, on average, informative”. In this study, the
authors create a penalty function minimizing which results in obtaining spread regions of implied ratings.
This approach was recovered by Jainming Kou and Simone Varotto in 2008 [5] with some modifications
and with statistical analysis to measure the goodness of the implied ratings in anticipating the agencies
down/upgrades. The same penalty-function approach is implemented by Fitch [7], with the difference
(among others) that the dataset is now composed by CDS spreads rather than bond spreads. Moody’s
MIR - Market Implied Ratings [9] provides three types of market implied ratings: bond-implied, CDSimplied and equity-implied. While we have already mentioned the latter, bond-implied and CDS-implied
MIR ratings are derived by filtering the data excluding extreme values, then calculating the median
spread for each agency rating class and taking boundaries between two classes by geometric means of
two contiguous median spreads. Standard & Poor’s MDS - Market Derived Signals [6] uses a linear
regression approach, with regressors: agency rating class, the industrial sector (for the corporate case),
the Document Type (the definition of what is a credit default) and others.
SOVEREIGN RATINGS IMPLIED BY COUPLED CDS-BOND MARKET DATA
3
In a recent paper [14] a proposal that uses unsupervised learning methods1 was presented, namely a
clustering algorithm based on an Hidden Markov Model, taking as inputs surviving probability curves
bootstrapped by CDS spreads.
As far as we are aware, all the published studies are mainly dedicated to the corporate sector with
some minor hints to sovereigns. Here we deal explicitly with the sovereign sector, firstly calculating
implied ratings with a standard penalty-function methodology over CDS (which we call, for simplicity,
Kou-Varotto (KV) approach), and then proposing and implementing a new methodology for calculating
market implied sovereign ratings (but portable to corporates as well). We then use some statistical
metrics to compare the performances of the two implied ratings sets.
3. sovereign market implied ratings: a CDS penalty-function analysis
In this section, we are going to briefly review the ideas of using a penalty-function to calculate implied
ratings from CDS spreads and then we present the results of such a calculation on a pool of 36 sovereign
countries for a 10-year time interval starting from 2004.
The basic assumption for this kind of non-structural approaches, mentioned in the previous section,
is that the agency ratings are considered on average informative, but with “errors”, due e.g. to delay of
the agencies in officially updating their opinion on the country’s ability to pay back its debt. Namely, in
this kind of approaches, the agency ratings are inputs to the calculation. Thus, the problem one faces
is the following: we have, for each trading day, a certain number of CDS spreads, let us say one for
each country we are considering2; attached to each country-spread there is a rating label (given by a
chosen agency, or by an average rating among all agencies); if agency ratings exactly captured the credit
risk inherent in the CDS spreads on that particular day, then the spreads of each rating class would be
perfectly separated along the CDS dimension: typically, this is not the case. We refer to figure 1 for an
example: you can see that AAA and AA rated countries (according to S&P’s) do not perfectly separate
along the CDS dimension, even if, on average, they do. The penalty-function approach aims to find a
clever way to draw a boundary in the spreads in order to give a new label (the implied ratings) on the
basis of that boundary, as clarified in figure 1.
The penalty function is derived from [4] and [5], and has the following expression (for AAA/AA
boundary)
N∑
NAA
AAA
1
1 ∑
+
+
penalty(b) =
(sAAA
−
b)
+
(b − sAA
(3.1)
j ) ,
NAAA j=1 j
NAA j=1
where:
• NAAA (NAA ) is the number of AAA (AA) rated countries,
• sAAA
(sAA
j
j ) is the spread of the j-th country among the AAA (AA) rated ones,
• b is the spread of the boundary that we are looking for,
• (·)+ denotes the positive part.
The penalty function increases its value whenever a AA rated country has a spread lower than b or a
AAA rated country trades at spreads higher than b. Minimizing 3.1 with respect to b gives the desired
spread of the boundary between AA implied and AAA implied rated spreads. What we are actually
doing is to choose the boundary that minimizes the “errors” of the agency ratings; the measure of the
error is simply the distance from the boundary itself.
The dataset we shall use comprehends a set of 36 countries3 covering all the rating classes; 5-year
maturity CDS spreads (being the more liquid ones) as given by Markit; long-term ratings according to
Standard & Poor’s; a time interval going from January, 1 st, 2004 to January, 14-th, 2014. In order for
statistical analysis to be more reliable, we map the S&P’s rating classes4 into 5 larger classes, according
to the following scheme:
AAA → AAA
1
See, e.g., [15].
Actually, for each country there are CDS for various tenors, but let us focus on one single maturity.
3See the appendix A for more details on the dataset.
4See, e.g., the S&P’s website www.standardandpoors.com for reference to S&P’s ratings classes and their meaning.
2
4
STEFANO MARMI, ALDO NASSIGH, AND DANIELE REGOLI
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0.006
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implied AA
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CDS5Y
RTG
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implied AAA
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Oct 06
Oct 07
Oct 08
Oct 09
Oct 10
time
● AA
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● AAA
●
Figure 1. snapshot of the 5-year maturity CDS spreads of AAA, AA rated countries
(according to S&P’s) in early October 2008, with a sketch of CDS implied rating construction for one day.
AA+,AA,AA- → AA
A+,A,A- → A
BBB+,BBB,BBB- → BBB
BB+,BB,BB-,B+,B,B-,CCC+,CCC,CCC-,CC,SD → BB
the last of which is necessary since we have few data for the single categories of lowest ratings.
The methodology is the following: for each day collect the spreads of the past 30 days (around 22
trading days); divide all the spreads according to the 5 rating classes; minimize the penalty function 3.1
for each couple of nearby ratings (AAA/AA, AA/A, …) thus finding 4 boundaries per day. The reason
for using data of the entire previous month is to prevent the implied ratings to be too volatile and too
connected to the daily movements of the market: Kou and Varotto in [5] collect data of two months
before the desired date. We believe that an accurate analysis of the best temporal window from which
collecting the data to calculate the daily boundary is worth doing, and we shall possibly deal with that
in future work.
Figure 2 shows the result of the calculation: you see the 4 barriers during the 10-year period. Of
course, the barriers move in time, reflecting the fact that there are some dynamical features common
to all the ratings, depending on common underlying scenarios. Indeed, the values of spreads are not
influenced by credit risk alone: market-wide liquidity situation, average risk aversion of investors, are
just some of the factors impacting on CDS premiums. From the plot, it is easy to see that before the
2008 crisis all the spreads were extremely low and rather stable, then you have a huge increase in all
the spreads, with very high correlation among the different classes; there is a peak in early 2009, then
a decrease in late 2009 and 2010, and then a new bump, nearly as high as the previous one, during the
so-called euro-zone sovereign crisis; then again a decrease in 2013. Notice that, after the triggering of the
2008 crisis, the spread barriers never revert to something comparable to the pre-crisis situation.
SOVEREIGN RATINGS IMPLIED BY COUPLED CDS-BOND MARKET DATA
5
AAA−AA
AA−A
A−BBB
BBB−BB
0.04
0.03
0.00
0.01
0.02
CDS5Y
0.05
0.06
KV barriers
2004
2006
2008
2010
2012
2014
time
Figure 2. time evolution of KV-implied rating boundaries
The core idea is that a crossing of one of these barriers by the CDS spread related to one country
detects precisely a change in the genuine credit risk part of the information encoded in spreads.
Once you have the boundaries, calculating the implied ratings for the countries is straightforward. As
an example, we plot the dynamics of the S&P’s and implied ratings for three countries, Italy, Spain and
Germany in figures 3, 4, 5 respectively5.
Some comments on the results shown in figures 3, 4, 5: firstly, it is clear that the KV implied ratings
anticipate all the downgrades, both for Italy and Spain; secondly, the anticipation horizon is quite long:
strong signals of downgrade starts between 1 and 2 years before the actual rating adjustment, sometimes
even more; KV implied ratings seem to be a little bit on the pessimistic side, particularly regarding the
recent years. These stylized features are common to the majority of the 36 countries analyzed. Moreover,
focusing on the German situation, we see that the CDS market was giving signals of downgrade for many
days in a row, both for the 2008 crisis and for the more recent euro-zone sovereign distress period, while
the CRAs didn’t change their excellent opinion on the creditworthiness of Germany.
4. CDS+bond based analysis
As we mentioned at the end of the previous section 3, KV analysis gives signals of downgrade for
Germany during the two main crisis of the last few years. Actually, even if German CDS spreads were
raising, resulting in downgrade market signals, we know that precisely in that periods, Germany, in
particular German bund, was the “safe haven” of the entire euro-zone. Indeed, as figure 6 shows, in
2011-2013, while the CDS spread was very high, the bond was actually very low. The very same pattern
was in place during the 2008 crisis as well. As was pointed out in [11], for sovereign countries the CDS
alone may not be able to capture enough information to assess credit risk. In [11] it is also argued that
taking into account both CDS and bonds should improve the ability of deducing from the market the
rating for countries. And that is precisely what we are going to investigate in what follows.
5The dataset for Italy starts in 2006 and not in 2004 because of a lack of data for the previous period.
6
STEFANO MARMI, ALDO NASSIGH, AND DANIELE REGOLI
kv implied ratings for ITALY
SnP
kv
AAA
AA
A
BBB
BB
2004
2006
2008
2010
2012
2014
time
Figure 3. Kou-Varotto implied rating together with historical S&P’s rating for Italy
kv implied ratings for SPAIN
SnP
kv
AAA
AA
A
BBB
BB
2004
2006
2008
2010
2012
time
Figure 4. KV implied rating together with historical S&P’s rating for Spain
2014
SOVEREIGN RATINGS IMPLIED BY COUPLED CDS-BOND MARKET DATA
7
kv implied ratings for Germany
AAA
SnP
kv
AA
2004
2006
2008
2010
2012
2014
time
Figure 5. KV implied rating together with historical S&P’s rating for Germany
BND5Y
−0.25
−0.50
−0.75
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time
Figure 6. plot of bond spread (upper) and CDS spread (lower) for Germany
RTG
●
AAA
8
STEFANO MARMI, ALDO NASSIGH, AND DANIELE REGOLI
Adding 5-year maturity bonds means adding one dimension to the input space: a single sovereign
state is represented no more by a single number (the CDS spread), but by a couple bond spread + CDS
spread. Thus, as time goes on, each country describes a trajectory in the bond-CDS plane. Now the
problem amounts to divide, on a daily basis, such a plane in regions, each corresponding to a different
implied rating. As in the one dimensional case we needed a prescription (the Kou-Varotto minimization
approach) by which separate the points into implied-rating classes, we need a prescription here as well.
One natural choice to classify points in multi dimensional spaces is to use the so called Support Vector
Machine technique [16, 17, 18, 19, 15, 20]. While we refer to the bibliography for a thorough description
of this well known approach of computer science, here we merely try and summarize the key points, for
sake of clarity and reader’s comprehension.
Suppose you have points on a plane, with coordinates xi , i = 1 . . . , N where N is the total number
of points, and suppose that each point comes together with a label yi ∈ {−1, 1}. We want to separate
these points with a line, in such a way that we divide the +1’s from the −1’s. The configuration can
be as easy as in the left panel of figure 7, which is called linearly separable: the +1’s and −1’s can be
exactly separated with a line. Actually, with an infinite number of lines. Among them, the Support
Vector Machine approach chooses the one that maximizes the margin as depicted in the same figure. In
this situation, the SVM is just a minimization procedure to find a unique line separating points on a
plane. Referring again to the left panel of figure 7 for the notation:
• (β0 , β) are the parameters of the separating hyperplane, with ∥β∥ = 1 (namely, β is the unit
vector normal to the hyperplane),
• x1 and x2 denote the closest points to the plane, the so called support vectors
• M is the half-margin
considering that f (xi ) = xTi β + β0 is precisely the (signed) distance of the point xi from the hyperplane
x : f (x) = 0, then, without loss of generality, we can take f (namely β, β0 ) such that
yi f (xi ) ≥ 0
is the distance of xi from the plane. Then, the problem of maximizing the margin looks like:
max
M
β,β0 ,∥β∥=1
subject to yi (β T xi + β0 ) ≥ M , ∀i
(4.1)
which is pretty self explanatory.
Actually, there a scale invariance in the problem: to see this, notice that one can replace the constraints
∥β∥ = 1, yi (β T xi + β0 ) ≥ M with
yi (β T xi + β0 ) ≥ M ∥β∥
(where β0 has been redefined appropriately); now notice that this constraint is invariant to rescaling of
the vector (β0 , β). Then, we can arbitrarily fix the norm ∥β∥. A smart choice is
1
.
M
Then, the problem (4.1) is equivalent to the following
∥β∥ =
min
β,β0
∥β∥2
2
subject to: yi (β T xi + β0 ) ≥ 1 , ∀i
(4.2)
In most cases, and for sure in our situation, data are not linearly separable. SVM prescribes a procedure
to deal with non-linearly separable data, as in the example of the right panel of figure 7: we still try to
maximize the margin, but we allow some points to be inside the margin or even on the “wrong” side of
the boundary line. These “errors” are weighted with the parameter ξi that is the distance by which they
enter inside the margin. The minimization problem to uniquely fix the linear boundary reads:

)
(
n
2
∑

∥β∥


+C
ξi ; C → cost parameter
min
β,β0
2
(4.3)
optimization
i=1


 subject to: y (β T x + β ) ≥ 1 − ξ , ξ ≥ 0 , i = 1 . . . n
i
i
0
i
i
SOVEREIGN RATINGS IMPLIED BY COUPLED CDS-BOND MARKET DATA
x2
yi = −1
9
yi = −1
M
M
ξ>0
x1
ξ>1
margin
ξ>1
yi = 1
xT β + β 0 = 0
yi = 1
xT β + β 0 = 0
Figure 7. Support Vector Machine separating plane: linearly separable (left) and nonlinearly separable (right) cases.
where C is an exogenous parameter that measures how much we are willing to make “errors” at all. The
minimization problem is then a trade off between the maximization of the margin and the minimization
of the cost we pay when making “errors”.
Actually, the real utilization of SVM is to use that boundary line to classify new data of which we know
x but we don’t know y. That’s precisely the philosophy behind all the Machine Learning algorithms: to
train the machine with some known data and to use it to classify unknown ones. We have instead to face
a slightly different problem: we want to separate points on a plane of which we know the classification
(agency ratings) but we also know that this classification contains “errors”, and we need a good way to
perform that separation.
Of course, SVM is not confined to work on a plane: SVM can deal with an arbitrary (finite, but in
some cases also infinite) number of dimensions. Moreover, SVM can deal with non-linear boundaries as
well, via so called kernel methods. In the present work we confine ourselves to linear boundaries and
to 2-dimensional analysis, but these are two features that we are going to investigate in future research:
adding to CDS spreads and bonds some extra information, and analyzing the possibility of implementing
some curved decision boundary.
The dataset we work with is the following: same 36 countries, ratings and CDS spreads as for the
previous section 3; 5-year maturity bond interest rate spreads over zero-interest curves, as calculated by
UniCredit risk management staff. The time interval is again January, 1st, 2004 → January, 14th, 2014.
The methodology for calculation is as follows:
• for each trading day, we collect CDS spreads and bond spreads for the 36 countries for 30 days
backward in time, resulting in approximately 750 input data per day,
• with these data we calculate the boundary lines for that day, with the SVM procedure,
• having the plane for that day divided into implied rating regions, it is easy to infer the implied
rating for each CDS-bond couple for that day.
All the calculations were done with R open software [21] and in particular with package e1071 [22] and
caret [23] for SVM analysis and cross-validation tuning respectively.
Actually, there are a couple of technicalities we must dwell on a while: one regarding the cost parameter
C, and the other referring to the multi-class nature of our problem. The cost parameter C must be fixed
a priori of the actual boundary calculation. Remember that fixing C means fixing the price we are
willing to pay for “misplaced” points on the plane. In the literature on SVM, one can find several
approaches regarding the parameter tuning of the support vector approach, among which we choose the
cross-validation. Namely, the routine does the following: she divides in 10 parts all the training data for
that day (∼ 750 points), then uses 9/10 of the data as training set on which she calculates the boundaries,
and the other 1/10 as a test set. This procedure is repeated for each 1/10 of the data set, and the overall
procedure is repeated for 10 times independently of one another. Then, the accuracy (i.e. how many
10
STEFANO MARMI, ALDO NASSIGH, AND DANIELE REGOLI
points of the test set are classified correctly by the trained boundaries), averaged over all these trials,
is calculated. This is done for every value of the cost parameter along the sequence (2−3 , 2−2 . . . 0, 2, 4).
Finally, the cost parameter that maximizes the accuracy is chosen. This procedure is run through every
ten training days, keeping the latest optimal cost for the other 9 days. The overall scheme results in a
further (this time statistical) optimization of the separation boundary.
Now with the multi-class issue: the SVM procedure we described above, refers to binary classification
problems, namely, problems where data belongs to one among 2 classes. Here, instead, we are dealing
with a 5-class classification. The routine we used adopts the so-called one-versus-one approach, which
runs like this: she calculates 10 linear boundaries, 1 for each couple of rating classes; then, each point on
the plane in assigned an implied rating by a voting scheme: the point is classified according to all the 10
couples, and the rating class that obtains more “votes” is the one assigned to that point. This procedure
determines, daily, the implied rating regions of the CDS-bond plane6.
Another minor technical remark is about the asymmetry of data: namely, for each day we have
(sometimes big) differences in the numbers of points for each class (namely, we have a lot of AAA countries
but few BB, see appendix A). Given that accuracy is the metric for cross-validation, unbalanced data
may result in unbalanced model selection. The solution to this problem is to assign a cost parameter C
different for each class, weighted on the basis of the size of the training dataset of each class. See, e.g.,
[16] for some details.
Unfortunately, showing the 2-dimensional boundaries evolving in time is neither illustrative nor easy
as in the one-dimensional case. We are going to show just a couple of daily snapshots, in order to give
an idea of how the SVM works. Some general comments regarding the two figures 8, 9:
• interestingly, you see that while the high ratings (AAA,AA) are separated from the others along
the SE-NW direction, the low ones tend to be separated along the orthogonal direction: this means
that while for high rated countries CDS and bond spreads tend to be negatively correlated, the
opposite holds for low rated countries,
• a comparison of the two plots clarifies the very idea of implied ratings and of SVM as well: in
fig. 9 you see that Romania and Turkey, despite being rated BB by Standard & Poor’s, have
actually quite low bond spreads, and not very high CDS spreads as well (comparable to BBB
values); SVM decides however to rate them BB, minimizing the errors; in fig. 8 on the other
hand, while Romania is still reclassified BB, Turkey is too mixed with AA guys, and classifying
it BB would have cost way too much: thus the calculation puts Turkey in the AA region. Indeed,
the market on those days whose trading Turkey debt issues and CDS at the level of most AA
countries.
As for the KV case, let us show the time evolution of implied ratings for Italy, Spain and Germany.
Figures 10, 11, 12 shows the results: historical S&P’s ratings, SVM implied ratings and, for comparison,
KV implied ratings. Regarding the issues we addressed for KV implied ratings, we see here that SVM
classification has at least two differences/improvements:
• it is more timely, meaning that the downgrading signals usually start nearer to the actual rating
change,
• it is more “optimistic”, meaning that tends to give higher implied ratings, in particular in the
recent period.
Regarding the German situation, we clearly see that adding the bund has the (expected) result of getting
rid of the downgrading signals both for the 2008 and for the euro-zone crisis.
5. statistical analysis: SVM vs. KV implied ratings
In the present section, we are going to present the results of some statistical tests done to measure the
goodness of our implied ratings, in comparison with the KV CDS-based approach.
6Another popular scheme in dealing with k-class classification is the one-versus-all, in which only k SVM boundaries
are calculated, and the classes are assigned according to the distance of a point from the k boundaries, on the basis of the
fact that the more you are far from a boundary the more likely your classification is right
SOVEREIGN RATINGS IMPLIED BY COUPLED CDS-BOND MARKET DATA
11
SVM classification plot
0.06
0.05
BB
romania
italy
BBB
turkey
spain
0.03
germany
A
SPREAD5Y
0.04
AA
0.02
AAA
0.01
malaysia
−1
0
1
norway
2
3
4
5
6
BND5Y
Figure 8. (part of the) bond-CDS plane on 2012-09-15: different colored regions denote
different implied ratings as calculated by the SVM procedure; the colored circles are the
data used to train the machine for this day. Colors of circles correspond to agency
(S&P’s) ratings according to: black-AAA, red-AA, green-A, blue-BBB, cyan-BB. Some
countries are indicated explicitly.
The usual tests to check the goodness of a classification are the ROC (Receiver Operator Characteristic)
curve and the Accuracy Ratio, with varying time horizon to test the predictive power as well. Both these
tools measure the ability of rating assignments essentially by considering the percentage of defaulters that
were assigned low ratings. Unfortunately, these are completely useless in our settings (and, in general,
for sovereigns), since the number of defaults of sovereign states, even if we considered longer period of
time and larger pool of states, is ridiculous in the perspective of having a statistical test. Thus, we are
obliged to abandon the idea of having a test of the absolute goodness of our implied ratings, and we check
their performance with respect to S&P’s.
5.1. accuracy. In order to check the accuracy of SVM implied ratings with respect to S&P’s, and to
compare it to that of KV implied ratings, we do the following: we take the implied ratings on each day
in the interval considered (2004, January 1st to 2014, January 14th) and we calculate the accuracy of
the SVM (and KV) implied ratings versus the agency rating of 1 year later. Then we plot the results of
accuracy day by day in figure 13. Recall that accuracy in a classification problem is simply given by
accuracy =
true positives + true negatives
;
entire population
namely, the fraction of right guesses7 made by the implied ratings.
In figures 14-16 other accuracy graphs are displayed, namely the daily accuracy of prediction for 3month, 6-month and 1 month horizon. It is pretty clear that in all the situations considered the SVM
implied ratings behave better than KV. Just to be clear, we can calculate the cumulate accuracy for the
7“Right” with respect to the (time-shifted) agency ratings.
12
STEFANO MARMI, ALDO NASSIGH, AND DANIELE REGOLI
SVM classification plot
romania
turkey
italy
A
0.03
BBB
spain
0.02
0.01
AA
slovenia
germany
0
1
2
AAA
SPREAD5Y
0.04
BB
0.05
3
4
5
BND5Y
Figure 9. (part of the) bond-CDS plane on 2013-09-12: different colored regions denote
different implied ratings as calculated by the SVM procedure; the colored circles are the
data used to train the machine for this day. Colors of circles correspond to agency
(S&P’s) ratings according to: black-AAA, red-AA, green-A, blue-BBB, cyan-BB. Some
countries are indicated explicitly.
SVM AAA’ AA’
A’ BBB’ BB’
KV AAA’ AA’
A’ BBB’ BB’
AAA 27562 3634 223
4
0
AAA 24604 6051
768
0
0
AA
AA
3828 9423 1531
496
53
3598 8270 2883
398
182
A
A
365 2640 8805 1506
175
136 3260 8009 1297
789
8 184 1871 8411
961
0 125 2219 6265 2826
BBB
BBB
BB
61
331
47 1252 5283
BB
0
13
808 1346 4807
Table I. cumulate contingency table for SVM (left) and KV (right) implied ratings for
1-month horizon (primed ratings stand for implied).
SVM AAA’ AA’
A’ BBB’ BB’
KV AAA’
AAA 25141 2490
61
0
0
AAA 22634
AA
3434 8550 1280
218
9
AA
2941
365 2641 7729 1151
2
136
A
A
BBB
8 301 1845 7182 1249
BBB
0
61
351
266 1787 3654
0
BB
BB
Table II. cumulate contingency table for SVM (left) and
for 1-year horizon (primed ratings stand for implied).
AA’
A’ BBB’ BB’
4586
472
0
0
7625 2698
171
56
3359 7473
841
79
119 1947 5952 2567
14
905 1267 3933
KV (right) implied ratings
entire period for the different horizons:
svm 1 month: ∼ 0.756 svm 3-month: ∼ 0.768
svm 6-month: ∼ 0.763
svm 1yr: ∼ 0.757
SOVEREIGN RATINGS IMPLIED BY COUPLED CDS-BOND MARKET DATA
13
svm vs kv implied ratings for ITALY
●
AAA
svm
SnP
kv
AA
A
BBB
BB
2008
2010
2012
2014
time
Figure 10. SVM implied ratings for Italy, in comparison with KV implied ratings and
together with the historical S&P’s.
svm vs kv implied ratings for SPAIN
●
AAA
svm
SnP
kv
AA
A
BBB
BB
2004
2006
2008
2010
2012
2014
time
Figure 11. SVM implied ratings for Spain, in comparison with KV implied ratings and
together with the historical S&P’s.
14
STEFANO MARMI, ALDO NASSIGH, AND DANIELE REGOLI
svm vs kv implied ratings for Germany
AAA
●
AA
2004
2006
2008
2010
2012
svm
SnP
kv
2014
time
Figure 12. SVM implied ratings for Germany, in comparison with KV implied ratings
and together with the historical S&P’s.
daily accuracy of prediction for 1yr horizon
0.8
accuracy
irs
SVM
KV
0.6
0.4
2004
2006
2008
2010
2012
2014
time
Figure 13. plot of daily accuracy of prediction for 1 year horizon, both for SVM and
KV implied ratings (irs).
SOVEREIGN RATINGS IMPLIED BY COUPLED CDS-BOND MARKET DATA
15
daily accuracy of prediction for 3−month horizon
1.0
0.8
accuracy
irs
SVM
KV
0.6
0.4
2004
2006
2008
2010
2012
2014
time
Figure 14. plot of daily accuracy of prediction for a 3-month horizon for SVM and KV
implied ratings.
kv 1 month: ∼ 0.665 kv 3-month: ∼ 0.670
kv 6-month: ∼ 0.677
kv 1yr: ∼ 0.686 ,
The best performance is given by SVM implied ratings over a 1 month predicting horizon. You can also
see the contingency tables for 1-year and 1-month horizon in tables I and II.
This analysis reflects two facts:
• SVM implied ratings are, on average, fairly better in matching future agency ratings with respect
to KV ones,
• SVM implied ratings (for the very same way in which they are concocted) tend to be pretty stable
around the agency ratings, resulting in an increasing accuracy the shorter the horizon.
As we shall see in section 5.3, this second point does not actually mean that the shorter term is the better
in predicting upgrades and downgrades of agency ratings. Namely, accuracy captures the information of
prediction in every situation, also when there is actually no upgrade or downgrade in the future. Some
kind of analysis which focuses mainly on predicting the agency ratings movements is necessary for a
thorough description of the goodness of implied ratings.
5.2. precision & recall. Another standard test we are going to show, is the precision & recall for each
class. For sake of clarity, we just give the cumulative precision and average recall over the 10-year interval
for each class, and we limit ourselves to the 1-month and 1-year horizons of prediction. The results are
summarized in tables III and IV
Let us remind what precision and recall are:
precision of class k =
recall of class k =
# true positives
= probability of being in class k given that the test was positive
# positives
# true positives
= probability of positive test given that the individual is in class k
# members of class k
16
STEFANO MARMI, ALDO NASSIGH, AND DANIELE REGOLI
daily accuracy of prediction for 6−month horizon
0.8
accuracy
irs
SVM
KV
0.6
0.4
2004
2006
2008
2010
2012
2014
time
Figure 15. plot of daily accuracy of prediction for a 6-month horizon for SVM and KV
implied ratings.
daily accuracy for 1 month horizon
1.0
0.8
accuracy
irs
SVM
KV
0.6
0.4
2004
2006
2008
2010
2012
2014
time
Figure 16. plot of daily accuracy of prediction for a 1-month horizon for SVM and KV
implied ratings
SOVEREIGN RATINGS IMPLIED BY COUPLED CDS-BOND MARKET DATA
17
horizon
precision (cumulate over 2004→2014) recall (cumulate over 2004→2014)
1 month AAA AA
A
BBB
BB
AAA AA
A
BBB
BB
svm
0.866 0.581 0.706 0.721
0.816
0.877 0.615 0.654 0.736 0.758
kv
0.868 0.467 0.545 0.673
0.559
0.783 0.539 0.594 0.548 0.689
Table III. precision & recall for the 5 classes, cumulate over the entire time interval
and with 1-month predicting horizon.
horizon precision (cumulate over 2004→2014) recall (cumulate over 2004→2014)
1 year
AAA AA
A
BBB
BB
AAA AA
A
BBB
BB
svm
0.867 0.597 0.6910 0.698 0.744 0.908 0.634 0.650 0.679 0.597
kv
0.880 0.486 0.554 0.723 0.593 0.817 0.565 0.629 0.562 0.643
Table IV. precision & recall for the 5 classes, cumulate over the entire time interval
and with 1-year predicting horizon.
horizon
F 1 (cumulate over 2004→2014)
1 month AAA AA
A
BBB
BB
svm
0.872 0.597 0.678 0.728 0.785
kv
0.823 0.500 0.568 0.604 0.617
Table V. F 1-score (as by formula (5.1)) for the 5 classes, cumulate over the entire time
interval and with 1-month predicting horizon.
horizon
F 1 (cumulate over
1 year
AAA AA
A
svm
0.887 0.615 0.670
kv
0.848 0.522 0.589
Table VI. F 1-score (as by formula (5.1)) for the
time interval and with 1-year predicting horizon.
2004→2014)
BBB
BB
0.687 0.662
0.633 0.617
5 classes, cumulate over the entire
The higher precision the fewer type I errors, the higher recall the fewer type II errors. Namely, with high
class k precision you can trust the test in signals of being in class k, with high class k recall you can be
quite sure that if a guy is in class k the test is going to get it right.
We can add as a final indicator the popular F 1-score, which is simply the geometric mean of precision
and recall, namely
precision × recall
F1 = 2 ×
,
(5.1)
precision + recall
and is a standard way of summarizing together precision and recall. The cumulate results for 1-month
horizon are in table V, while in figure 17 we plot the daily values of F 1 for all the 5 classes. In table VI
and figure 18 you can see the same indicator but for a 1-year predicting horizon.
As it is apparent from table III, both precision and recall (and thus F 1 as well) are quite high for
all the rating classes, and the SVM implied classification performs systematically better, with the (very
slight) exception of the precision of AAA.
5.3. gap-conditioned analysis. In this section we want to address the capacity of sovereign implied
ratings to predict agency rating adjustments. Namely, as was mentioned at the end of section 5.1, the
metrics seen so far do not discriminate predictions of stability from predictions of down/upgrades. While
predicting stable ratings is of course important, a focus on rating adjustments is needed to understand the
goodness of implied ratings. For the statistical test analysis of the present section, we take inspiration by
Moody’s report [9], where the authors do a similar thing for corporates (with their bond-implied ratings).
18
STEFANO MARMI, ALDO NASSIGH, AND DANIELE REGOLI
F1 − AAA
F1 − AA
1.0
1.00
irs
0.8
0.6
irs
0.75
SVM
KV
SVM
0.50
KV
0.25
0.4
2004
2006
2008
2010
2012
2014
2004
2006
F1 − A
2008
2010
2012
2014
F1 − BBB
1.00
1.00
irs
0.75
irs
0.75
SVM
0.50
KV
0.25
SVM
0.50
KV
0.25
2004
2006
2008
2010
2012
2014
2004
2006
2008
2010
2012
2014
F1 − BB
1.0
irs
0.8
0.6
SVM
0.4
KV
0.2
2004
2006
2008
2010
2012
2014
Figure 17. F 1-score for 1-month horizon for the 5 rating classes.
F1 − AAA
F1 − AA
1.0
1.00
irs
0.8
0.6
irs
0.75
SVM
KV
SVM
0.50
KV
0.25
0.4
2004
2006
2008
2010
2012
2014
2004
2006
F1 − A
2008
2010
2012
2014
F1 − BBB
1.00
1.00
irs
0.75
irs
0.75
SVM
0.50
KV
0.25
SVM
0.50
KV
0.25
2004
2006
2008
2010
2012
2014
2004
2006
2008
2010
2012
F1 − BB
1.0
irs
0.8
SVM
0.6
KV
0.4
2004
2006
2008
2010
2012
2014
Figure 18. F 1-score for 1-year horizon for the 5 rating classes.
2014
SOVEREIGN RATINGS IMPLIED BY COUPLED CDS-BOND MARKET DATA
19
In figure 19 you see a graph displaying predictive power of SVM implied ratings. The methodology
is the following: we take the 36 countries at the beginning of each month from 2004, January 1st to 6
months before our end date, which is 2014, January 14; we calculate, for each country in each of these
days, the rating gap, namely:
rating gap = iRTG − RTG
were we have mapped rating to integers according to
AAA → 1, AA → 2, . . . ,
BB → 5 ;
thus, a positive rating gap means that, on that particular day, the SVM procedure assigns a rating worse
than the agency does. Then we go and look to the situation 6 months later and we count how many
countries have been downgraded/upgraded/untouched by S&P’s, conditioned to their rating gap (of 6
months before). We aggregate the data for all the countries and the entire time interval and plot them in
a bar graph. Each bar corresponds to a rating gap and counts all the countries that, being in that rating
gap class, are subject to an (agency) upgrade (green), a downgrade (blue), or are stable (red). Notice
that the bars show only fraction values, namely the count is normalized with respect to the number of
events for each class.
If the implied ratings are to lead the agency ratings (i.e. to predict them) than we would expect the
following: a very big red bar in the 0-gap position; as we move rightward, an increasing blue region
(a worse implied rating ’implies’ an increasing fraction of future downgrades); as we move leftward, an
increasing green region (a better implied ratings ’implies’ an increasing fraction of future upgrades). As
we can clearly see, while for gap ≥ 0 we can indeed conclude that implied ratings lead the agency ratings,
we cannot say the same for negative rating gaps.
The same calculation is done for a 1 year horizon: the SVM performance is here even better, which
means that, when focusing on rating adjustments, the short term horizon is not the optimal choice for
prediction.
Table VII and plot 20 show the same results, but with absolute numbers. You can see that the vast
majority of events is in the region: gap = 0 – change = stable. Actually, notice that the bars corresponding
to gap -4, -3, 3 have far too few events to be statistically meaningful: it is better to consider only the
bars with gaps form -2 to 2. Moreover, it is apparent that the upgrade events are actually just a tiny
fraction of the overall number, both in the 6-month and 1 year scenario, thus resulting in a statistically
poor ensemble for making any prevision regarding the upgrades.
down stable up
down stable up
-4
0
3
0
-4
0
3
0
-3
-3
0
15
0
0
15
0
-2
-2
0
26
0
0
26
0
-1
0
382 19
-1
6
331 33
0
63
2500 22
0
115
2276 38
1
44
291
0
1
84
227
0
2
15
21
0
2
23
13
0
3
0
0
3
0
0
3
3
Table VII. 6-month and 1 year SVM gap-conditioned contingency tables
In figure 21 it is displayed the same bar graph, obtained with the very same methodology described
above, with rating gap calculated with Kou-Varotto implied ratings, while VIII and 22 are the corresponding tables and absolute counting bar graphs, respectively. A comparison shows clearly that, as far
as downgrades are concerned, SVM implied ratings perform slightly better than KV.
6. Conclusions
Credit ratings have a crucial role in many aspects of modern financial markets, even more so after
the 2008 mortgage crisis (when CRAs have been strongly criticized for their ‘opinions’ on mortgagebacked securities) and in particular euro-zone crisis. Given that rating agencies are quite slow in updating
20
STEFANO MARMI, ALDO NASSIGH, AND DANIELE REGOLI
6−month rating change conditioned to SVM gap
1.00
count (%)
0.75
down
stable
up
0.50
0.25
0.00
−3
−2
−1
0
rating gap
1
2
3
1yr rating change conditioned to SVM gap
1.00
count (%)
0.75
down
stable
up
0.50
0.25
0.00
−4
−3
−2
−1
0
1
2
3
rating gap
Figure 19. 6-month and 1 year predictive power: on x axis the gap between SVM
implied rating and the corresponding historical one (positive gap means implied rating
is worse than agency rating); the bars show the fraction of stable/upward/downward
rating change after 6 months-1 year.
SOVEREIGN RATINGS IMPLIED BY COUPLED CDS-BOND MARKET DATA
21
counting of SVM gap−conditioned 6−month horizon rating change
2000
count
1500
down
stable
up
1000
500
0
−3
−2
−1
0
rating gap
1
2
3
counting of SVM gap−conditioned 1yr horizon rating change
2500
2000
1500
count
down
stable
up
1000
500
0
−4
−3
−2
−1
0
1
2
3
rating gap
Figure 20. 6-month and 1 year absolute counting for rating change conditioned to SVM
rating gap on the entire time interval.
22
STEFANO MARMI, ALDO NASSIGH, AND DANIELE REGOLI
6−month rating change conditioned to KV gap
1.00
count (%)
0.75
down
stable
up
0.50
0.25
0.00
−3
−2
−1
0
rating gap
1
2
3
1yr rating change conditioned to KV gap
1.00
count (%)
0.75
down
stable
up
0.50
0.25
0.00
−3
−2
−1
0
rating gap
1
2
3
Figure 21. 6-month and 1 year predictive power: on x axis the gap between KV implied rating and the corresponding historical one (positive gap means implied rating is
worse than agency rating); the bars show the fraction of stable/upward/downward rating
change after 6 months-1 year.
SOVEREIGN RATINGS IMPLIED BY COUPLED CDS-BOND MARKET DATA
23
counting of KV gap−conditioned 6−month horizon rating change
2000
count
1500
down
stable
up
1000
500
0
−3
−2
−1
0
rating gap
1
2
3
counting of KV gap−conditioned 1yr horizon rating change
2000
count
1500
down
stable
up
1000
500
0
−3
−2
−1
0
rating gap
1
2
3
Figure 22. 6-month and 1 year absolute counting for rating change conditioned to KV
rating gap on the entire time interval.
24
STEFANO MARMI, ALDO NASSIGH, AND DANIELE REGOLI
down stable up
down stable up
-3
0
1
0
-3
0
1
0
-2
0
45
0
-2
0
42
0
-1
1
420 30
-1
2
361 53
23
2200 11
69
2021 18
0
0
1
64
513
0
1
104
432
0
2
32
57
0
2
51
32
0
3
5
2
0
3
5
2
0
Table VIII. 6-month and 1 year KV gap-conditioned contingency tables
their credit assessments – particularly for sovereign countries – market implied ratings have become
increasingly necessary in providing timely early warnings for institutions/investors interested in higher
frequency information about creditworthiness of counterparts. In this paper, we focused our study on
sovereign market implied ratings, whilst the literature is mainly devoted to the corporate sector. We
proposed a methodology for deriving implied ratings by market data, exploiting both CDS and bond
market informations. As far as we know, this methodology, even if inspired by the works in [4] and [5], is
entirely new in the respect of using two different market sources together and also in the implementation
of Machine Learning techniques for this purpose.
We performed the calculation of the implied ratings for a panel of 36 countries during a 10-year period
starting in 2004; we calculated both “standard” implied ratings (namely, using only CDS spreads as input
information and implementing [4]-[5] technique) and the new ones. The results show that both calculations
provide quite good early warnings, both for anticipating upgrades/downgrades and for predicting stable
ratings. However, the implied ratings derived with the new methodology perform systematically better
in all respects.
This is intended as preliminary work on this subject, further analysis must be put in place in order to
improve and generalize the idea here proposed and to test the results here obtained with larger samples
and possibly other statistical metrics.
Appendix A. data details
In the following, some detailed information about the dataset used for statistical analysis is presented.
We acknowledge UniCredit for providing all the data used in our calculations, in particular we thank
Dr. Andrea Basile and the entire Group of Risk Methodologies and Architecture.
The 36 countries considered are Australia, Austria, Belgium, Bulgaria, Canada, Croatia, Czech Republic, Cyprus, Denmark, Finland, France, Germany, Greece, Hong Kong, Hungary, Ireland, Italy, Japan,
Malaysia, Netherlands, New Zealand, Norway, Poland, Portugal, Romania, Russia, Singapore, Slovenia,
Slovak Republic, Spain, Sweden, Swiss, Turkey, Venezuela, UK, US.
Credit Default Swaps. CDS data are from Markit Corporation. They are updated to 14-01-2014, with
quoted maturities in {0.5, 1, 2, 3, 4, 5, 7, 10, 15, 20, 30} years, among which we used the most liquid one,
namely 5-year. The Debt Tier considered is Foreign Currency Debt (Domestic Debt CDS quotations are
much more rare), and currencies for considered debt are dollars for all countries except for US who has
euros denominated contracts.
Bonds. Bond data are constructed starting from bond prices by Bloomberg Corporation. The spread
derivation goes through the construction of two curves: the zero curve, or risk-free curve, one for each
currency area (namely, we have a zero curve for each currency by which the bond contracts are denominated), and the zero-coupon curve based on treasuries (of course, one curve for each country considered).
The actual data used are the spreads between the treasury (zero-coupon) yield and the risk-free yield
(of the corresponding currency). The zero curves are built essentially starting from quotations of basic
derivatives on currency exchange rates. On theoretical grounds put forward in [24], the zero-coupon
curves for treasuries are bootstrapped via the the standard risk-free procedures, even if treasuries are in
fact defaultable. As for CDS, the most liquid maturity (5-year) only has been used in calculation.
SOVEREIGN RATINGS IMPLIED BY COUPLED CDS-BOND MARKET DATA
25
Agency ratings. Agency ratings are by Standard & Poor’s. We use the Long Term Foreign Currency
rating assessment, remapped into 5 classes as described in section 3. S&P’s is the agency typically more
timely in updating/reviewing its rating assessments.
On average along the period analyzed, roughly 39% of the countries considered are rated AAA by
S&P’s8, 20% AA, 17% A, 14% BBB and 10% BB.
As a last remark, notice that we have not complete data for all the countries on the entire period
considered (01-01-2004 → 14-01-2014). This refers almost completely to the bond dataset, where countries
like Swiss, Italy, Singapore, New Zealand, Bulgaria, Portugal, Malaysia, Venezuela and Romania, have
yield curves starting after 2004 (typically around 2007). This, of course, does not influence at all the
analysis of this paper, it only amounts to have a shorter time period of analysis for some of the countries.
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Scuola Normale Superiore and C.N.R.S. UMI 3483 Laboratorio Fibonacci, Pisa, Italy
E-mail address: [email protected]
UniCredit Group, Milano and Università Commerciale Luigi Bocconi – Milan, Italy
E-mail address: [email protected]
Scuola Normale Superiore, Pisa, Italy
E-mail address: [email protected]