Reconciling credit correlations

The Journal of Risk Model Validation (47–64)
Volume 4/Number 2, Summer 2010
Reconciling credit correlations
Andrew Chernih
University of New South Wales, Sydney, NSW 2052, Australia;
email: [email protected]
Luc Henrard
Catholic University of Louvain, School of Management, Place des Doyens 1,
B-1348 Louvain-la-Neuve, Belgium; email: [email protected]
Steven Vanduffel
Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium;
email: [email protected]
The credit crisis has resulted in a new impetus for regulators in analyzing
the framework for determining regulatory capital requirements; in particular, the assessment of credit risk will be challenged. Confronted with a
lack of default statistics, it is common for industry practitioners to apply
a financial approach known as Merton’s model of the firm, which also
underpins modern solvency standards such as Basel II and Solvency II.
However, while Merton’s theory is an academic beauty, its implementation
does not make full use of available default statistics but, instead, relies on
the concept of so-called asset correlations. We study the different estimates
used for asset correlations that have appeared in the literature and analyze
to what extent these estimates are in line with each other, with available
default statistics and with our own findings. Our results are the same as
most of those found in the literature but deviate from the results reported by
some major software providers as well as from the Basel II and Solvency II
figures. We offer several explanations to reconcile these differences and
point to several other features that should not be overlooked when building
credit portfolio models.
1 INTRODUCTION AND CONTEXT
Over the past decade, banks and insurance companies have made considerable investments in the design and implementation of risk management systems. However, the
financial crisis has increased the pressure on financial institutions and regulators to
challenge and potentially revise all concepts and models used; credit risk models
are particularly subject to such a critical review process. As far as credit risk is
The authors would like to thank Alan Pitts and Karl Rappl (UBS), Jon Frye (Federal Reserve Bank
of Chicago), Ivan Goethals (ING) and Bruno de Cleen (Rabobank) for helpful discussions and
comments on an earlier draft.
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concerned, it is clear that different obligors usually operate in related socio-economic
environments so that, at least to some extent, their assets are impacted similarly;
this points to a positive dependence between the default events. Therefore, we
cannot readily resort to the law of large numbers to claim that for sufficiently large
portfolios, the actual defaults will always be “as expected” and hence no capital will
be needed. On the contrary, capital is necessary to absorb adverse deviations from
what is expected during the time horizon at hand.
To further elaborate on the need for capital, let us consider, for a given time
horizon, an infinitely large homogeneous portfolio, ie, a portfolio with an infinite
number of obligors having identical characteristics. In this case, all of the single
default probabilities, denoted by qsingle , and pairwise default probabilities, denoted
by qpair , will be similar across the different obligors involved. Furthermore, the socalled exposures-at-default (ie, the maximum losses in the case of a default) and
loss-given-defaults (ie, the percentage of effective loss upon default) will be equal
to each other. The quantities “exposures-at-default” and “loss-given-defaults” will
be denoted by EAD and LGD, respectively. By additionally assuming that EAD and
LGD are fixed numbers rather than random variables, it is easily shown that the
standard deviation of the portfolio loss S, considered as a fraction of the total amount
at risk (also known as the “portfolio unexpected loss”), can be expressed as:
Stdev(S) = ρ D · Var(L)
(1)
Here, L is the individual obligor loss random variable (also expressed as a fraction),
whose variance is given by:
Var(L) = LGD2 · qsingle · (1 − qsingle )
(2)
In Equation (1), ρ D ≥ 0 is the default correlation (for the given time horizon)
between the different obligors:
ρD =
2
qpair − qsingle
qsingle · (1 − qsingle )
(3)
Looking at (1), (2) and (3) together, we observe that Stdev(S) depends on the loss
given defaults (LGD), the individual default probabilities (qsingle ) and the pairwise
default probabilities (qpair ). While the process for estimating qsingle and LGD may be
fairly under control, the intrinsic lack of sufficient default statistics puts a burden on
the estimation of qpair and consequently casts doubt on the accuracy of the estimate
for Stdev(S) and, ultimately, the required capital.
Confronted with the apparent lack of default data, regulators and financial
institutions resort to other techniques to derive the pairwise default probabilities
qpair . In particular (see, for instance, Deloitte and Touche (2004)), they often rely on
“Merton’s model of the firm”, which uses the intuitive idea of a default modeled as an
adverse event triggered by an asset value moving in the wrong direction. Technically
The Journal of Risk Model Validation
Volume 4/Number 2, Summer 2010
Reconciling credit correlations
speaking, Merton’s model amounts to assuming that, for the time horizon under
consideration, the asset (log) returns are multivariate normally distributed, implying
that a default for the ith borrower (where i = 1, 2, . . . , n) occurs when the change
in asset value Ni drops below a certain threshold. Hence, in the context of our full
homogeneous portfolio, the multivariate normal character of the asset returns implies
that Ni can be expressed as:
(4)
N i = ρ A M + 1 − ρ A i
so that:
qsingle = Pr(Ni < c)
= Pr( ρ A M + 1 − ρ A i < c)
(5)
Here, M and the i (i = 1, 2, . . . , n) are independent standard normally distributed
random variables representing the common systemic factor (also called the state
of the economy) and the idiosyncratic risks, respectively. Furthermore, ρ A ≥ 0 is
the (asset) correlation coefficient, c is the constant threshold value and (·) is the
distribution function of a standard normally distributed random variable, with inverse
−1 (·). Of course, joint defaults will be driven by the correlated asset returns, and it
is easily shown that:
qpair = ρ A (−1 (qsingle ), −1 (qsingle ))
(6)
where ρ A (· , ·) is used to denote the distribution function of a bivariate standard
normal random pair with correlation coefficient ρ A ; see Crouhy et al (2000) for more
details. Note that in Equation (5), the probability qsingle (= (c)) does not depend on
the precise state of the economy M and is therefore unconditional. On the other hand,
Equation (5) can also be used to define conditional probabilities, which we denote
by qsingle (m), as follows:
c − ρAm
qsingle (m) = Pr i < 1 − ρA
−1
(qsingle ) − ρ A m
(7)
=
1 − ρA
The conditional probability qsingle (m) is the probability that there will be a default
given that the (future) state of the economy M is known to be equal to m.
It is important to observe that under Merton’s firm value model, Equation (6)
provides an explicit relation between the (unconditional) parameters qsingle , qpair
and ρ A , which enables the “translation” of default correlations into “equivalent”
asset correlations and vice versa. Indeed, if default statistics have been used to
derive estimates for qsingle and qpair (and thus also the default correlation ρ D ), then
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Equation (6) can be used to extract the implied asset correlation ρ A . On the other
hand, if qsingle and ρ A are available, then we can use Equation (6) to derive qpair
and hence, by Equation (3), also the implied default correlation ρ D . In this respect,
we remark that in the literature on credit correlations it is now common to use asset
correlations (rather than default correlations) to discuss and compare the different
findings.
Merton’s firm value model can also be used to derive the loss distribution of the
infinitely large homogeneous portfolio exactly; consequently, the maximal loss at a
given confidence level p, namely the “portfolio value-at-risk”, denoted by VaRp (S),
can be determined as well. Indeed, by using the fact that conditional on the state of
the economy M the default events are independent, it follows from the law of large
numbers that, essentially:
VaRp (S) = VaRp (E(S | M))
= VaRp (LGD · qsingle (M))
(8)
Thus we find that:
−1 (qsingle ) + ρ A · −1 (p)
VaRp (S) = LGD · 1 − ρA
(9)
Formula (9) can be traced back to Vasicek (2002) and is widely used. In fact,
many credit risk portfolio models rely on (variants of) Merton’s model of the firm
as summarized above. For example, Basel II relies on (9) to determine the required
capital that banks need for their credit portfolios; see Basel Committee on Banking
Supervision (2006, pp. 64–66) and Basel Committee on Banking Supervision
(2005). Also, the upcoming Solvency II framework is likely to use formula (9),
with a minimum level of 50% for the asset correlations, for determining capital
charges to cover for exposure to reinsurance or derivative counterparts. We refer
to the Committee of Insurance and Occupational Pension Supervisors (2008) for an
overview of the technical Solvency II guidelines, and to Doff (2008) for a detailed
critical analysis of the whole Solvency II framework.
We would also like to point out that under the stated assumptions of an infinitely
large homogeneous portfolio, Equation (8) holds generally, so that in such instances
capital formulae will always be determined essentially by the properties of the
random variable qsingle (M). In fact, under Merton’s paradigm, qsingle (·) is determined
by (7) whereas M is a standard normal random variable representing the uncertain
future state of the economy, and this gives rise to formula (9) for VaRp (S). However,
if other assumptions are made with regards to qsingle (·) and M, then obviously other
capital formulae will result. For example, let us assume that defaults indeed occur
according to Merton’s paradigm but that, in addition, we are able to predict with
certainty the state of the economy, ie, it is known ex ante that M will be equal to
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Reconciling credit correlations
some m∗ . Then formula (8) will still hold, but now we find that:
VaRp (S) = VaRp (LGD · qsingle (m∗ ))
= LGD · qsingle (m∗ )
(10)
so that VaRp (S) is nothing more than a (conditional) expected loss in this case.
The example makes clear that value-at-risk and capital calculations are intimately
connected with the design of the rating system used to generate default probabilities.
If a so-called through-the-cycle rating system is put in place that generates (unconditional) probabilities qsingle as in (5), where M is unknown, then formula (9) is
appropriate for determining value-at-risk. In contrast, if we have in place a perfect
point-in-time rating system where M is known with certainty, then formula (10) is
correct. In fact, if capital were taken to be the difference between the value-at-risk
and expected loss, then no capital would even be needed because “everything will
evolve as expected”; see also Cornaglia and Morone (2009) for some recent work on
this topic. In this paper, we use Merton’s paradigm as the basis and the results of the
analysis have to be understood in this context.
While it is fair to say that Merton’s idea has enhanced the understanding and
modeling of credit portfolio risk in significant ways, it needs to be stressed that, in
practice, its implementation relies on asset value data, which is not directly observable but can only be derived (through a series of other models and assumptions) from
equity value data. Indeed, although under the Merton paradigm default statistics are
likely to be used to derive estimates for the single default probabilities qsingle , the
estimation of the pairwise probabilities qpair (see (6)), and thus also the portfolio
unexpected loss Stdev(S) as well as the portfolio value-at-risk VaRp (S) (see (8)), is
not backed fully by default statistics but, rather, relies on asset value data; see also
Crouhy et al (2000).
It is important to keep in mind that, even though the wider availability of asset
value data along with Merton’s appealing theory may provide some feelings of
comfort, ultimately the performance of any credit default risk model needs to be
assessed using default statistics. In this respect we agree with Frye (2008), who
stated that “many naïve risk managers rely on asset correlations without checking the
results”. It would therefore be of interest to review the different estimates used for
asset correlations that have been mentioned in the literature, analyze to what extent
these estimates agree with each other, and try to offer explanations for any observed
differences. At the same time, some of these explanations may shed light on other
features that should not be overlooked when building credit portfolio models (which
involves considering more than just correlations), and we provide some insight with
regard to the impact that these may have on the results.
As far as we know, this is the first study that provides a global overview of
findings in the literature on the level of asset correlations, and we will compare
the results of our study, which is based on a large sample of monthly asset value
data, with a series of other studies. These other studies can be divided roughly into
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two categories. One kind of study uses observed default data to calculate single and
pairwise default frequencies, from which default correlations can then be derived
directly; this category of research includes Gordy (2000), Frey and McNeil (2003),
Dietsch and Petey (2004), Jobst and de Servigny (2005) and de Servigny and Renault
(2002). The second category of work involves the use of asset or equity values to
obtain estimates for the asset correlations; papers in this category include Duellmann
et al (2008), Lopez (2002), Pitts (2004), Zeng and Zhang (2001) and Akhavein et al
(2005). The last two papers report findings from Moody’s KMV (MKMV) and Fitch
Ratings, respectively.
We find that the results of our study are broadly in line with the most of those
in the literature, and several plausible reasons have already been documented to
explain observed differences. On the other hand, all these results deviate from what
has been reported by MKMV and Fitch Ratings, as well as from the Solvency II
and Basel II figures. We offer some potential explanations in an attempt to reconcile
these differences. Firstly, we provide evidence that credit risk modeling is subject to
significant modeling risk, which implies that some stakeholders will, for reasons
of conservatism, simply prefer to use higher correlations than what has actually
been observed. In particular, this could explain the high level of correlations used in
Basel II and Solvency II. Secondly, we analyze two sources of dependence that might
exist in a credit portfolio and that are typically overlooked by modelers, namely,
dependent loss-given-defaults and the presence of group effects. This may also
explain, to some extent, why some software providers, rating agencies and regulators
use higher-than-observed correlations.
The paper is structured as follows. Section 2 gives an overview of results in
the existing literature, whereas Section 3 describes the results we have obtained.
Section 4 discusses all of the various results and offers some potential explanations
to reconcile the observed differences. Finally, Section 5 contains some concluding
remarks.
2 ASSET CORRELATIONS FROM THE LITERATURE
2.1 Asset correlations using default data
In Table 1 (see page 53) we report the asset correlations from a variety of studies
that made use of observed default data. When the equivalent asset correlations were
not explicitly mentioned in the study, we used Equation (6) to extract these from the
reported default correlations.
Note that all (inherent) default correlations are unconditional and that the abovementioned studies have used a variety of methods to estimate these. For example,
both Gordy (2000) and Cespedes (2002) appear to have first estimated the standard
deviation of the portfolio loss corresponding to the population at hand, and then
relied on Equation (1) to derive the default correlation. Other studies, such as
Hamerle et al (2003a), use a straightforward model-free technique; for more details,
see Lucas (1995).
The Journal of Risk Model Validation
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Reconciling credit correlations
TABLE 1 Asset correlations derived from default data.
Source study
Default data source
Results (%)
Gordy (2000, Table 2)
Cespedes (2002)
Hamerle et al (2003a)
Hamerle et al (2003b, Table 1)
Frey and McNeil (2003, Table 1)
Dietsch and Petey (2004)
Jobst and de Servigny (2005)
Duellmann and Scheule (2003)
Jakubik (2006)
S&P
Moody’s
Unknown
S&P 1982–1999
S&P 1981–2000
Coface 1994–2001
S&P 1981–2003
DB 1987–2000
BF 1988–2003
1.5–12.5
10
Max 2.3
0.4–6.04
6.5–6.9–9.1
0.12–10.72
4.7–14.6
0.5–6.4
5.7
S&P: Standard and Poor’s; DB: Deutsche Bundesbank; BF: Bank of Finland.
TABLE 2 Asset correlations derived from asset value data.
Source study
Duellmann et al (2008)
Zeng and Zhang (2001)
Akhavein et al (2005)
Lopez (2002)
de Servigny and Renault (2002)
Asset data source
Results (%)
MKMV Credit Monitor
MKMV source
Equity
MKMV Portfolio Manager
Equity
10.2
9.46–19.98
20.92–24.09
11.25
6
We mention that, in addition to the studies listed in Table 1, Hamerle et al (2003a)
used the same data as in Boegelein et al (2002), comprising default data from
Canada, France, Germany, the UK, Italy, Japan, South Korea, Singapore, Sweden
and the US. There is also a study by Vassiliev (2006), which uses default data from
UBS to calculate asset correlations that “are in the range reported in external studies
(eg, Dietsch and Petey (2004))”.
From Table 1 we can conclude that the different results, while using different estimation techniques to cope with default statistics from the US, Canada, Switzerland,
France, Germany, Finland, the UK, Italy, Japan, South Korea and Singapore that
cover different data periods as well, yield broadly consistent correlation estimates
in the range of 1%–12%, approximately. Note that the default data used in these
studies was typically available at a one-year horizon, meaning that the implied asset
correlations correspond to a one-year time horizon in these instances as well.
2.2 Asset correlations using asset value data
Using asset value data as input, we can estimate asset correlations directly in a
straightforward way (and also move on to default correlations if necessary). Several
studies exist that have made use of asset value data from various sources; the results
are summarized in Table 2.
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Note that, in contrast to studies based on default data, most asset value studies use
a weekly or monthly horizon to calculate asset returns and derive asset correlations.
Duellmann et al (2008), however, calculated two-year asset correlations using rolling
24-month time windows. Since their study used only eight years of asset return
data (which means four distinct 24-month periods), this may be insufficient for
correlation estimates. Akhavein et al (2005) used equity values and a proprietary
software to report five-year correlations. Zeng and Zhang (2001) do not release full
details concerning how they built their asset correlation model or the data they used,
and their model is designed to be used in conjunction with the MKMV Portfolio
Manager software. Lopez (2002) also did not disclose details of the asset data used,
only that MKMV Portfolio Manager was used for the analysis. Presumably, MKMV
Portfolio Manager is based on the same raw data (from MKMV Credit Monitor) that
we used in our analysis, but there is no formal guarantee of this. We also point out
that de Servigny and Renault (2002) effectively utilized equity correlations to obtain
an average correlation of 6%. As well as the studies listed in Table 2, we mention
that Basel II also uses asset correlations ranging from 8% to 24%, with the exact
value depending on individual firm characteristics such as rating and asset size, to
determine the required capital that banks need for their corporate credit portfolios;
see the Basel Committee on Banking Supervision (2006, pp. 64–66). Finally, the
upcoming Solvency II framework is likely to set a minimum level of 50% for the
asset correlations in determining capital charges to cover for exposure to reinsurance
or derivative counterparts; see the Committee of Insurance and Occupational Pension
Supervisors (2008).
The results in Table 2 suggest that, compared to using default data, the use of
asset value data to estimate asset correlations appears to give rise to slightly higher
estimates in general; see Frye (2005) for comments in a similar vein. Gordy and
Heitfield (2002) have shown that using the specifications of Merton’s model in the
statistical procedure for estimating asset correlations based on default data is likely
to yield higher results in general; see also Miu and Ozdemir (2007) for related work
in the context of estimating default probabilities. It seems that using model-free
methods to estimate asset correlations based on default data may be subject to some
downward bias, hence explaining why the asset correlations from Table 1 tend to
be lower. Nevertheless, such downward bias is only really correct and meaningful
if the real default data process is (broadly) in line with Merton’s theory, which is,
owing to the scarcity of observed default data, difficult to prove in a statistically
meaningful way. In other words, if real defaults do not occur according to the Merton
mechanism, then there is no real guarantee that the estimates in Table 1 are too low;
we should also keep in mind that model-free procedures are by nature more robust
with respect to model specification. Finally, we remark that the studies in Zeng and
Zhang (2001) and Akhavein et al (2005) give significantly larger results than any
other study reported so far.
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Reconciling credit correlations
3 OUR ASSET CORRELATIONS
3.1 The data used
The source of the asset value data used in this study is MKMV Credit Monitor.
Several other papers, such as Pitts (2004) and Duellmann et al (2008), have used the
same raw data source. In both of these papers it is stated that this asset data should be
corrected for the impact of corporate actions and potential data errors. Accordingly,
the data was cleaned to remove outliers, asset values were adjusted for debt issues
and buybacks, and those months for which asset value data was not available were
removed from consideration.
The monthly returns were then calculated as the ratio of the ending (market)
asset value minus the value of liabilities issued during that month to the starting
(market) asset value. This means that for each of the 20,144 companies, we had up
to 107 months of return data available over the time period from March 1998 to
March 2007. To the best of our knowledge, this is the largest sample that has ever
been used for an asset correlation study.
Whenever a pair of companies had at least 40 months’ worth of data in common,
a correlation was calculated. The companies were aggregated into 336 clusters based
on asset size, activity sector, probability of default and world region. There is some
evidence from previous studies that these are factors that may differentiate asset
correlation; see Lopez (2002), Duellmann and Scheule (2003) and Dietsch and Petey
(2004), among others.
3.2 Our main findings
For our 336 asset clusters, the average intra-cluster asset correlation was 11.1% (this
was obtained by calculating for each cluster the average of all pairwise correlations
that exist within the cluster and then averaging over all clusters); the average intercluster asset correlation was 6.3% (obtained by calculating for each pair of clusters
the average of all pairwise correlations that exist between two assets from the two
different clusters, and then taking the average over all combinations of two different
clusters). A graph of the inter-cluster correlation, grouped by asset size band and
client rating, is shown in Figure 1 (on page 56).
As expected, the correlations increase with asset size (because the greater the
asset size the more systemic the obligor’s risk profile typically becomes). Moreover,
in line with common expectations, we see that correlations decrease with increasing
probability of default (because, in general, there is an inverse relation between default
probability and asset size).
4 DISCUSSION
The results we have obtained from using the monthly asset return data are consistent
with most of the values found in the existing literature, and it is fair to say that results
in the literature are in general very closely aligned (with the exception of Akhavein
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FIGURE 1 Asset correlations by S&P rating and asset size band.
AAA–BB+
BB–B-
B+
B–C
8.00%
7.00%
Average asset correlation
56
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
Very small
Small
Medium
Asset size bands
Large
Within each asset size band, the four default probability groups are plotted from left to right in
decreasing order of S&P rating (ie, increasing order of default probability).
et al (2005), who reports equity correlations as a substitute for asset correlations
using return data spanning a five-year horizon, and Zeng and Zhang (2001)). Thus,
on the one hand, there appears to be a consensus in the literature on the range of
values for asset correlations; but, on the other hand, the question arises as to why
Fitch Ratings and MKMV, as well as Basel II and Solvency II, use significantly
different values. We now proceed to offer a number of possible explanations. Some
of these explanations may shed light on other features that should not be overlooked
when building credit portfolio models (correlations are not the only component to
take into account), and we provide some insight into the impact that these factors
may have on the results.
4.1 Correlations do not measure dependence fully
First, let us remark that any given set of assumptions regarding the loss-givendefaults, exposures-at-default, single default probabilities and asset or default correlations will never be complete enough to estimate the portfolio loss distribution
with certainty. Indeed, since correlations are related to individual and pairwise but
not multiple default probabilities, they do not provide a full picture of the dependence
that exists within a portfolio. In fact, all joint default probabilities need to be determined when estimating a credit loss distribution in general and the portfolio valueat-risk in particular. Unfortunately, however, even estimating a triple-wise default
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probability based on default data is a cumbersome exercise; so, in a credit context,
obtaining a full picture of dependence is a far-from-trivial task, and this problem has
a negative impact on the potential accuracy of the estimated loss distribution. This is
the reason why most companies make an implicit or explicit assumption regarding
the dependence structure; the assumption of multivariate normally distributed asset
returns in Merton’s model of the firm is just an (influential) example of such practice.
In this respect, we point out that it is even perfectly possible to construct models that
all preserve a given Stdev(S) while giving very different results for the portfolio
value-at-risk VaRp (S).
Let us now assume that the normalized asset returns Ni evolve as follows:
Ni = M · IM<d + −1 ((d) + (i ) · (−d)) · (1 − IM<d )
(11)
where IM<d is an indicator random variable and d is a threshold value reflecting
“bad times”. Note that while the different Ni are still normally distributed, they
are no longer multivariate normally distributed. Loosely speaking, the asset returns
are fully dependent (ie, exhibit maximal “tail correlation”) when the market is in a
deep recession, and they are unrelated otherwise. Then, whenever p > (−d) >
1 − qsingle holds, we will find that the portfolio value-at-risk at the confidence
level p, namely VaRp (S), is exactly equal to the maximal loss that can possibly
be incurred, ie, the loss when all obligors default. For example, let us consider
a homogeneous portfolio with qsingle = 1.5%, LGD = 100% and ρ A = 7%. Then,
using Monte Carlo simulation, we find that (−d) is approximately equal to 99%.
Hence, when p > 99%, under the model specification (11) for the asset returns we
have that VaRp (S) = 100%, whereas under the (Merton) specification (4) we obtain,
by formula (9), that VaRp (S) = 6.12%. Thus, while the parameter values for LGD,
EAD, qsingle and ρ A are exactly the same in both models, the results produced are
substantially different; see also Chernih et al (2008) for more information in the
same vein.
Confronted with significant intrinsic model error, some stakeholders may, for
reasons of conservatism, prefer to use higher correlations than what is actually
observed. This could, in particular, explain the high level of correlations used in
Basel II and Solvency II.
4.2 Effect of dependent loss-given-defaults
Furthermore, even the calculation of the portfolio unexpected loss is significantly
impacted by (explicit or implicit) modeling assumptions. Indeed, the computation
of portfolio unexpected loss is dependent on whether the loss-given-defaults are
deterministic or stochastic, the dependence between them, and possible dependence
between default events and loss-given-defaults. We will focus on the issue of
dependent loss-given-defaults.
To this end, let us consider again an infinitely large homogeneous portfolio.
However, while we still assume that the exposures-at-default are fixed, we will
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consider three different methods to deal with the loss-given-defaults. The first
model assumes stochastic dependent loss-given-defaults, the second model assumes
stochastic independent loss-given-defaults, and the third model assumes deterministic loss-given-defaults. It is very likely that the first (and most sophisticated) model
leads to the most accurate description of the portfolio risk. Indeed, different lossgiven-defaults are likely to be positively related to each other, because they are driven
by asset value processes that are most likely positively correlated as well.
All other things being equal, using the second or third model for loss-givendefaults will result in a lower estimate for the portfolio unexpected loss, Stdev(S),
than would be obtained from the first model. Alternatively, to achieve the best
estimate of the unexpected loss (defined as the result from the first model), the other
approaches will need to use higher asset correlations than those used in the first
model. This naturally leads to the question: how much higher?
From Dhaene et al (2006) we have the following equation for the loss correlation ρ L between individual random credit losses:
ρL =
A+B
Var(L)
(12)
where:
2
]ρ LGD · Var(LGD)
A = [ρ D · qsingle · (1 − qsingle ) + qsingle
(13)
B = ρ D · qsingle · (1 − qsingle ) · E 2 (LGD)
(14)
and:
Here, ρ LGD denotes the correlation between the loss-given-defaults for two distinct
credit risks, and E(LGD) is the expected value of the random loss-given-default
while Var(LGD) is its variance. Furthermore, the variance of an individual loss,
Var(L), is now given by:
Var(L) = E 2 (LGD) · qsingle · (1 − qsingle ) + qsingle · Var(LGD)
(15)
For the infinitely large portfolio as described above, it can be shown that the
portfolio unexpected loss expressed as a fraction of the total amount at risk equals:
Stdev(S) = ρ L · Var(L)
(16)
We can calculate Stdev(S) using the first model and then calculate what the required
asset correlation should be for the second and third models in order to attain the same
Stdev(S). Note that (16) reduces to (1) when ρ LGD = 0, meaning that for an infinitely
large portfolio it makes absolutely no difference whether loss-given-defaults are
stochastic and independent or deterministic. Hence the required asset correlations
for the second and third models will be the same.
We calculated the “corrected” asset correlations for various values of qsingle ,
2
σ (LGD) and ρ LGD . The expected value E(LGD) was fixed at 50%. Eight results are
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TABLE 3 Corrected asset correlations, for given values of qsingle , ρ LGD , Var(LGD)
and original asset correlation, that are needed to keep the same portfolio unexpected loss, assuming an infinitely large portfolio of obligors with ρ LGD = 0.
qsingle (%)
ρ LGD (%)
Var(LGD) (%)
Original asset
correlation (%)
Required asset
correlation (%)
25
100
25
100
25
100
25
100
25
25
4.2
4.2
25
25
4.2
4.2
13.96
13.96
13.96
13.96
8.45
8.45
8.45
8.45
16.84
23.32
14.48
15.94
16.88
37.53
9.93
14.18
0.21
0.21
0.21
0.21
9.75
9.75
9.75
9.75
presented in Table 3. For example, the first row represents a portfolio of obligors with
a qsingle of 0.21%, a ρ LGD of 25% and a σ 2 (LGD) of 25%. The original (estimated)
asset correlation is 13.96%, but an asset correlation of 16.84% needs to be used to
keep the same value of Stdev(S).
From Table 3 we observe that the relative change from the true asset correlation
to the corrected asset correlation (to account for dependent loss-given-defaults
implicitly) can be tremendous.
The effect of dependent loss-given-defaults may also explain the possible need to
use higher asset correlations in portfolio models. Often, though it might seem that
asset correlations are being measured, these are in fact used as input for calculating
the overall credit loss distribution. Thus, while best estimates of asset correlations are
typically in the region of 0%–10%, obtaining best estimates for credit loss distributions might require the use of higher asset correlations (to account for other sources
of dependencies). For example, in the MKMV Portfolio Manager software, the
loss-given-defaults appear to always be modeled as mutually independent random
variables, which may explain a need for upward adjustment of the originally derived
asset correlations to prevent potential underestimation of the portfolio unexpected
loss.
4.3 Group effects
Another kind of dependence that can exist in a portfolio and that is frequently
overlooked by modelers arises from group effects. A credit risk portfolio often
contains individual policies that have strong legal or economical ties. A conglomerate
(mother company) may be composed of different legal entities (daughter companies),
and a default of the former may lead to default of all of the latter. In fact, such
dependence is often taken into consideration when assessing the default probabilities
within a group of related companies; because the default of a daughter company
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may be prevented by the mother company, in the presence of group effects, daughter
companies might be assigned better ratings (and hence lower default probabilities)
than they would have received as stand-alone entities. Unfortunately, while such
group effects are often considered in the modeling of individual default probabilities,
they are almost never taken into account when modeling the dependence, and this can
lead to underestimation of the risk. Indeed, Vanduffel et al (2008) used comonotonic
theory (see Vanduffel (2005)) to extend a celebrated actuarial collective risk model
(see Panjer (1981)), known in the industry as CreditRisk+ , to account for such
group effects. Their numerical examples showed that such group effects can have
a significant impact on the loss distribution, especially in the tail.
Group effects are likely to increase the need for capital, but most credit risk
models fall short of capturing such effects. Therefore some stakeholders may opt
to consider the effect implicitly, by artificially increasing the levels of the asset
correlations.
4.4 Effect of the horizon
The correlation estimates reported in the literature do not necessarily all correspond
to the same risk horizon, and this may create bias when comparing the results.
Indeed, a priori, it is not obvious that observed asset returns involve the same
correlation across different horizons. While this would be true when asset returns
are multivariate normally distributed, it is by no means a general truth. Also, default
correlations may vary depending on the length of the horizon used. The effect of
the horizon has been studied for equity correlations (see, for example, Koyluoglu
et al (2003)) but less so for asset correlations. For the most part this is due to data
limitations. Default data is generally only available to use at a one- or five-year level.
In Jobst and de Servigny (2005), although default correlations were observed
to increase with increasing horizon (using one-, three- and five-year time periods),
the probability of defaults also increased, thus keeping the asset correlation broadly
constant. However, in de Servigny and Renault (2002) some evidence for increasing
asset correlations was found. Moving from a one-year time period to a longer time
period gives inconclusive results.
Using asset returns requires the use of shorter time periods, owing to less data
being available. Given that the general advice is to use at least 50 values to estimate
a correlation, using annual returns is impossible with the number of years of asset
return data available. Some commonly used alternatives are to calculate either
weekly or monthly asset returns and then use these as annual asset correlations.
Some studies, such as Duellmann et al (2008), use rolling-window time periods
to calculate correlations; that is, overlapping time periods are used. However, this
approach does not increase the effective dimension of the calculations. Using rolling
24-month time periods with eight years of data may lead to more observations, but
consecutive observations are now built on almost identical data, differing by only one
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month at the beginning and one month at the end: there are still merely four distinct
24-month periods to work with.
Our study was based on monthly asset returns, and the results suggest that using
monthly asset correlations as a proxy for asset correlations gives values that are in
line with default correlations observed from default data. Generally speaking, we
believe that the statistics used to parameterize the model should be aligned with the
risk horizon chosen. The impact of the horizon on asset correlations is certainly a
topic that requires further research.
4.5 Effect of the rating system
We recall from the introduction that rating systems and capital frameworks should
be aligned in principle. Indeed, in the context of an infinitely large portfolio,
Equation (8) holds more generally and reveals the close connection between the
portfolio value-at-risk VaRp (S) and the stochastic default probability qsingle (M).
Rating systems may differ in the way m → qsingle (m) is modeled, as well as with
respect to the modeled stochasticity of M. For example, under a perfect point-intime rating system, the stochasticity of M vanishes, so that no deviations around
the expected losses will be observed (for an infinitely large portfolio) and hence no
capital is needed to account for possible shocks (as it will be accounted for in the
provisions). In fact, formula (9) is only fully appropriate when Merton’s model of
the firm has been used to generate the (unconditional) probabilities qsingle .
Financial institutions often rely on internal rating systems to determine default
probabilities and on (Merton’s) formula (9) to determine capital requirements. As
the underpinning frameworks do not comply with each other per se, practitioners
may for reasons of conservativeness opt for artificially increasing the levels of the
asset correlations used in formula (9). This may also contribute to explaining why
Basel II and Solvency II are using higher-than-observed asset correlations, though it
still does not seem to provide a satisfactory explanation for the higher levels of asset
correlations used by Moody’s KMV.
5 FINAL REMARKS
Observed default data is a priori the best source to use for estimating single and
pairwise default frequencies; default correlations can then also be derived. However,
confronted with a lack of sufficient default data, banks, insurance companies and
software providers often resort to equity or asset correlations. Such correlations are
then transformed into default correlations by means of the so-called Merton model of
the firm. While the additional sophistication and the availability of asset value data
may provide some feelings of comfort, this comes at the cost of significant model
risk. Indeed, asset data itself is not observable but needs to be derived from equity
data using option models; see Crouhy et al (2000).
A discussion on which kind of asset correlations should be used and in how much
detail they need to be modeled is, in our view, a bit off-topic. Credit risk portfolio
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models are by nature subject to tremendous model risk anyway, and correlations are
only one piece of a very difficult puzzle, which unfortunately can never be fully
completed. In this respect, we believe that complex models by no means guarantee
more accurate results. For example, in view of the earlier considerations, it seems fair
to say that the added value a sophisticated multi-factor model for asset correlations
brings to the accuracy of capital calculations is questionable.
In our opinion, credit portfolio models should be kept as simple and transparent
as possible, capturing essential drivers such as correlation, dependent loss-givendefaults and group effects in a straightforward and consistent way, and making use of
default (loss) statistics as much as possible (including the use of externally available
default data). A candidate credit portfolio model that appears to meet several of
these criteria is the (one-factor) CreditRisk+ model, introduced by Crédit Suisse
and known in actuarial circles as an example of a collective risk model. Indeed,
the natural parameterization of this model is based on default statistics, calculations
can be done analytically, and features such as group dependence and dependent lossgiven-defaults can be readily accounted for; see, for instance, Vanduffel et al (2008)
and Vandendorpe et al (2008) for further information.
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