BANCO
DE
MÉXICO
THE IMPACT OF DEFAULT
CORRELATION ON
CONCENTRATION RISK IN BANK
LOAN PORTFOLIOS:
Constructing a concentration risk ratio and
adapting the CyRCE model to measure credit
risk
June 2002
THE IMPACT OF DEFAULT CORRELATION ON
CONCENTRATION RISK IN BANK LOAN PORTFOLIOS:
Constructing a concentration risk ratio and adapting the CyRCE model
to measure credit risk
Javier Márquez Diez-Canedo1
Risk Analysis and Special Projects Manager
Banco de México
June 2002
Summary
The first version of the CyRCE model stated that default correlation between
debtors increases concentration risk. This paper examines how the
correlation affects concentration risk by reformulating the general CyRCE
model so that the Rayleigh coefficient measures loss variance. Our analysis
is based on the concept of an equivalent correlation and shows that the
resulting variance is a convex combination of the maximum possible
concentration and loan portfolio concentration measured using the Herfindahl
and Hirschman index, in which the weigh is the said equivalent correlation.
This entails constructing a concentration risk ratio for a loan portfolio and
therefore analyzing the segments of it.
Besides allowing for an analysis of the correlation’s impact, this new model is
simpler and easier to implement than the former one, resulting in more
accurate individual limits, as shown in the first version’s example.
1
The author would like to thank Maribell Rojas Garduño, Rosa Emilia Ojeda, and Alberto
Romero for their assistance.
I. INTRODUCTION
Although the original CyRCE model is mathematically correct, putting it into practice is
complicated, as the change in the variable used means that the default variancecovariance matrix “M” must be factorized using its square root “S”; in other words: M =
STS. The problem is that this representation of S is not unique and requires great
numerical effort. Furthermore, although re-sizing the original loan vector resulting from the
change in variable G = SF clearly implies that concentration by number of loans (number
concentration) is not necessarily the same as concentration risk, it is difficult to measure
and visualize how the correlation affects number concentration and how concentration risk
is generated in the loan portfolio.
In this paper we derive the CyRCE model using once more the Rayleigh ratio to obtain
simpler ratios that are easier to manage in practice. Using the same analytical structure as
the original one, expressions are derived for VaR, capital adequacy, and individual limits.
The model is also extended to allow for an analysis of the segmented portfolio so as to
obtain differentiated individual limits and capital requirements per segment as well as
expressions indicating each segment’s contribution to the portfolio’s total risk.
Model parsimony provides a convenient environment for analyzing the impact of default
correlation on credit risk in general and concentration in particular. The analysis enables a
concentration risk ratio to be built that is very useful for analyzing a loan portfolio’s risk or
that of any of its segments. The results are illustrated by applying the new model to the data
of the same numerical example as in the original version.
II. A GENERAL CREDIT RISK MODEL
Consider a loan portfolio in which all loans have different default probabilities and are also
correlated. Let us assume that the default probability distribution can be
characterized by the median and the variance. The “ expected probabilities of
default π” vector a n d t h e corresponding “variance-covariance M” are known. The
amount at risk ratio is as follows:
√
By multiplying and dividing
by
, and then dividing the result by the value of the
portfolio
the capital adequacy ratio is obtained as follows:
̅
√
̅
where
is a measure of the standard deviation of losses and
̅
√
is the weighted average probability of default by relative loan size. We therefore obtain the
level of concentration and individual loan limit by applying theorem 5.1 of the original version2,
as follows:
(
̅
)
In fact, inequalities (1.2) and (1.5) are exactly the same as those obtained in the original
version, except that the loss variance is now measured using the “Rayleigh Coefficient”. Note
that the Rayleigh coefficient is a ratio of the loan vector norm resized using the
variance-covariance matrix and the original loan vector norm. It is easy to verify reduction
to simple one-dimensional case when default probabilities are equal or independent 3
for all debtors.
As indicated in the original version, the resizing of the loan vector using the variancecovariance matrix implies that number concentration is not the only factor that should be
borne in mind. In other words, a group of small loans with big default variances and high
correlations can represent a bigger risk than a small group of large loans with small variance
and default probabilities that are also not very correlated either with each other or with other
loans in the portfolio. Total loss variance √
is broken down in the default probabilities
variance effect, measured by the Rayleigh Coefficient, and the concentration effect,
measured by H(F).
2
3
See J. Márquez. Capital Adequacy and Credit Risk. Banxico Research document No. 2002-4
π = p1 and M = p(1-p)I.
III. The equivalent correlation, its impact on concentration risks, and
Concentration Risk Index
To examine the way in which credit risk increases when correlation increases, let us consider
the case in which probabilities of default are the same for all loans along with their
correlation coefficients; in other words, when the variance-covariance matrix can be stated as
follows:
(
)
(
)
When the distribution associated with individual loan defaults is Bernoulli ( ):
(
)
Applying (1.1) to this simple case, the expression for VaR is:
{
√
√
}
In the ratio stated above, the loss variance is broken down into two elements. The first refers
to the Bernoulli
variance while the concentration effect is represented as follows:
Note how this expression represents a convex combination between
of a fully
concentrated portfolio and the concentration index of the original loan portfolio
. The
portfolio thus behaves like that where debtors are independent but with an H’ concentration
that increases according to correlation coefficient “ ”. As a result,
might be considered a
correlation corrected concentration index. If debtors display default independence and there
is clearly no correlation
. On the other hand, if debtor defaults are
perfect and positively correlated
then
. This enables us to say that
is indeed a concentration risk index.
According to the aforementioned, a default probability and correlation coefficient “ ” can
be found for any loan portfolio risk level measured by VaR according to (1.1), such that:
[
Solving for the correlation coefficient we obtain:
]
(
)
(
)
[
]
[
]
The above expression tells us that the equivalent correlation coefficient for all portfolio loans
is a ratio divided by variance differences. In the numerator, it is the difference between the
loan portfolio loss variance and the “F” loan portfolio loss variance if all of the loans
had the same default probability “p” and were independent. In the denominator, it is the
difference between the loss variance of a fully concentrated loan portfolio and the loss
variance of the “F” loan portfolio if all of the loans had the same default probability
“p” and were independent. For practical purposes, if the loan portfolio’s weighted average
is taken as a probability of default (1.4), the implied correlation coefficient (3.6) can be
derived and the concentration risk index can be estimated accordingly (3.4).
EXAMPLE 3.1
To illustrate the previous results, consider the group of loans in the previous example; Table
6.1.
Table 6.1
N° of
loans
s
1
2
3
4
5
6
TOTAL
GRADE
TOTAL
C
D
E
F
G
$3,138 $5,320 $1,800 $1,933
$358 $22,805
$3,204 $5,765 $5,042 $2,317 $1,090 $30,994
$4,831 $20,239 $15,411 $2,411 $2,652 $45,544
$4,912
$2,598 $4,929 $12,439
$5,435
$6,467 $11,902
$6,480
$6,480
$12,456 $11,376 $21,520 $31,324 $22,253 $9,259 $21,976 $130,164
A
$4,728
$7,728
B
$5,528
$5,848
Table 7.1
Grade
Probability of Median (p)
default (%)
Std Dev. [p(1-p)]
A
B
C
D
E
F
G
1.65
3.00 5.00 7.50 10.00 15.00 30.00
12.74 17.06 21.79 26.34 30.00 35.71 45.83
Because of its size, the M covariance matrix was divided into three blocks4:
[
]
Assuming normality, the portfolio value at risk or amount at risk with a confidence level of 5%
is:
√
=
Now calculating:
√
̅
Recalling that
√
, the capital adequacy condition is:
̅
√
Assuming K = $60,000, such that
by calculating the individual limit as a
percentage of the value of the portfolio, from equation (1.5), we obtain:
4
These blocks are shown in Tables B.2 to B.7 of Appendix B.
(
̅
)
(
)
That is:
There are only two loans that exceed this limit (Table 7.2). Note that the level is exactly the
same as that obtained in the other version of the model.
Let us now examine the impact of the correlation in the concentration through the equivalent
correlation (3.6):
[
]
[
]
As a result, the concentration risk index is:
The portfolio in this example is a very poor one because most of the loans carry high default
probabilities. Adding the effect of the correlation on concentration we find that the unexpected
loss is √
compared with √
for loans with default
independence; in other words, the correlation duplicates the standard deviation of the loss
distribution of uncorrelated loans. As the analysis shows, the correlation effect results in a
concentration risk
four times greater than number concentration
,
such that the 43% capitalization requirement of the portfolio containing correlated loans is much
higher than the 27% required under the assumption of independence.§
Table 7.2
Probability of
default
Grade
A
F
Percentage
4,728
3,204
4,912
3.6%
2.5%
3.8%
1
2
3
C
C
0.0165
0.05
0.05
4
5
D
D
0.075
0.075
5,320
20,239
4.1%
15.5%
6
F
0.15
1,933
1.5%
7
8
9
F
G
B
0.15
0.3
0.03
2,598
1,090
5,528
2.0%
0.8%
4.2%
10
C
0.05
3,138
2.4%
11
12
13
C
0.05
0.1
0.1
4,831
5,042
15,411
3.7%
3.9%
11.8%
14
15
F
0.15
0.3
2,411
358
1.9%
0.3%
6,467
7,728
5.0%
5.9%
E
E
G
16
17
G
A
0.3
0.0165
18
B
0.03
5,848
4.5%
19
20
21
22
23
C
D
E
F
G
G
G
0.05
0.075
0.1
0.15
0.3
5,435
5,765
1,800
2,317
2,652
4.2%
4.4%
1.4%
1.8%
2.0%
0.3
0.3
4,929
6,480
3.8%
5.0%
24
25
130,164
EXAMPLE 3.2
As previously mentioned, number concentration is not the only factor that should be
considered. To illustrate this, consider a portfolio containing only one loan graded A:
Table 7.3
N° of
loans
1
GRADE
A
$130,164
PROBABILITY
STANDARD
OF DEFAULT
DEVIATION (%)
(%)
1.65
12.74
Since it concerns a portfolio with only one loan the
normality and choosing a confidence level of 5%,
(
index is equal to one, assuming
we obtain:
)
√
Assuming that the economic capital is $60,000, then the capitalization ratio is:
Considering the loan portfolio from example 3.1., the VaR of this portfolio is:
√ ̅
{ ̅
̅ √
[
where
}
]
and
Assuming the same economic capital, the capitalization ratio is:
If there were no correlation,
, then
. Consider the case where
in
other words, the case in which there is only one loan. The VaR of this portfolio would thus be,
{ ̅
√ ̅
̅ }
In other words, using this model, the VaR is almost three times greater than the VaR of the
portfolio described in Table 7.3 with only one loan. The difference derives from the
probability of default, as while one portfolio contains only one loan with a
probability of 0.0165, the other portfolio has a weighted probability of 0.1089 that is
almost 7 times the probability of the “A” grade loan.
If the correlation were unitary,
, then
, regardless of the concentration index in
numbers, and once again we find that the portfolio’s probability of default has all the weight.
7.3 HANDLING DIFFERENT SIZES OF CONCENTRATION
Generally speaking, banks classify their loan portfolios into “segments” based on some
practical classification criteria, such as geography, industry, and products. Another criterion is
to separate consumer loans by borrowers’ income level or corporate loans by the size of the
business, the classification being linked to the way in which banks do business.
In the case of credit risk in general and concentration risk in particular the aim is to adopt a
different criterion. As mentioned previously, one of the toughest problems is determining ex
ante, potentially dangerous concentrations, which may have nothing to do with the bank’s
organizational structure. Fortunately, the model that is developed allows for a totally
arbitrary segmentation of the portfolio such that it can be examined from various angles
and enables segments with a potentially risky concentration to be determined. This in turn
allows limits for each segment to be distinguished and implications in terms of capital
adequacy to be assessed. From this point on, the words class, segment and concentration
size will be used synonymously.
7.3.1. ANALIZING INDIVIDUAL SEGMENTS
Assuming that is randomly divided into classes,
, where
is a vector
that contains the balances of the loans belonging to the ..th segment. The expected
probabilities of the default vector and variance-covariance matrix are now divided as
follows:
a) π =
Partition of the probabilities of default vector where
“ ” is the probabilities of default vector of segment i; i
= 1,2,3,....,h
b) The variance-covariance matrix is divided as follows:
[
]
Each sub matrix
corresponds to the idiosyncratic variance-covariance matrix of group “i”
and has a size
; where Ni is the number of loans in the segment. All of these
matrixes are positive, defined the same as
and matrixes “ ” contain the covariances of
the probabilities of default between loans in group “i” and loans in group “j”.
Let
∑
be the value of the portfolio associated with “i”, and ∑
, where “ ” is the percentage of capital assigned to segment “i”,
Proceeding
as before, the inequality of value at risk for each class is:
.
√
[
. Let
] ∑
In the equation above it should be noted that matrix “ ” has the following shape:
[
]
Each matrix “ ” only takes into account correlations between the probabilities of default of
loans belonging to group “i” and those of other groups but eliminates correlations between
other groups with no direct impact on the group in question. Another way of writing the
Value at Risk inequality is:
∑
√
{
}
Dividing both sides by the value of portfolio Vi, we obtain:
̅
(
∑
)
{
where
√
√
}
.
It is interesting to note that the concentration level now includes a correction related to the
default correlation with loans in other segments (the second symbol to the right of the
inequality). This concurs with intuition, as a high default correlation with loans in other
segments indicates that less concentration in group “i” can be tolerated.
Using the same argument as in 7.2, individual limits
segment:
(
̅
are obtained for loans in each
∑
)
{
}
As with expression (7.6), the Rayleigh coefficient results in:
{(
̅
∑
)
{
}
}
Note that the difference versus the ratio (7.5) is that in (7.13) an adjustment is made for the
correlation between the default probabilities on loans in segment “i” and the loans in other
segments represented by the term
∑
{
∑
}
{
}
Clearly, when this term is positive, as is usually the case, the effect is to reduce limits on
individual segments. In sum, it shows how individual limits differentiated by segment which
depend on three factors are obtained, namely:
1. The segment’s idiosyncratic capital adequacy ratio:
̅
(
)
2. The correction for correlation with other segments:
∑
{
∑
}
{
}
3. The Rayleigh Coefficient, explicit in 1 and 2, or the characteristic
value greater than M.
7.3.2. CAPITAL ADEQUACY FOR A SEGMENTED PORTFOLIO
When including an analysis of individual segments of the portfolio, two things should be
considered. Firstly, the relative weights of each segment of the portfolio must not alter the
results obtained for the non-segmented portfolio. Secondly, an activity property that permits
the capital requirements of individual loan segments to be added in order to obtain the
requirement for the portfolio as a whole must be maintained. This can be achieved as follows:
Let
√
∑
√
and:
√
where
y∑
dividing (7.16) by
. It is easy to verify that
, an analysis similar to the previous section results in:
̅
∑
√
{
}
Solving for
we obtain
̅
(
)
∑
{
}
and for Theorem 6.1,
(
̅
∑
)
{
}
Equation (7.17) establishes capital adequacy for individual segments, (7.18) establishes a
concentration limit for the segment, and (7.19) provides the ratio for individual limits. Using the
Rayleigh coefficient and the results of Theorem 6.1, expressions for all of the above can be
obtained, only in terms of characteristic values.
Note that (7.17) and the following expressions are obtained from “ ⁄ ”, such that the
weight of the segments in the portfolio is not taken into account. Therefore, a simple sum of
terms can be misleading in terms of capital adequacy for the segmented portfolio. Therefore,
∑
let
. Then, by construction, if (7.17) is satisfied for all segments,
ensures capital adequacy for the portfolio.
Once more, the analysis produces some simple ratios from which capital adequacy can be
derived. The expressions obtained can be used as normative instruments for determining
individual limits, changes in the portfolio’s composition and/or adjustments to capital
requirements in order to maintain adequacy in the event of a change in loan default behavior
in one or all segments.
EXAMPLE 7.2.
To illustrate these results, the portfolio shown in Table 6.1 of previous examples will be used.
This portfolio was divided into three groups. The loan vector, the default probabilities, and the
variance-covariance matrix were divided as follows:
[
]

, is the idiosyncratic variance-covariance matrix of the first
group, which in this example consists of eight loans: A1, C2,
C4, D1, D3, F1, F4 and G2. 5

is the idiosyncratic variance-covariance matrix of the second
group. In this example, it is comprised of the following eight
loans: B1, C1, C3, E2, E3, F3, G1 and G5.

is the idiosyncratic variance-covariance matrix of the third
group, which in this example comprises the nine remaining
loans: A2, B2, C5, D2, E1, F2, G3, G4 and G6.

is the matrix of co-variances between the loans in
groups one and two. Likewise,
is the covariance matrix
between the loans in the first and second groups, and
is the matrix of co-variances between loans in the second and
third groups.
Note that for each segment the shape of matrix
[
]
is:
[
]
[
]
The first step in applying the results is to calculate the value of each segment, its
corresponding Herfindahl Hirschman index, and the corresponding capital allocation. These
numbers are summarized in the following table:
Table 7.3
Segment i
1
2
3
Vi
H(Fi)
$ 44,024
$ 43,186
$ 42,954
0.2613
0.2008
0.1293
yi
Ki
0.3382 $ 20,293
0.3318 $ 19,907
0.33 $ 19,800
As for (7.15), the parameters which allow the sum of adjusted VaRs is:
√
∑
√
By calculating with (7.16), using a confidence level of 5% and assuming normality, we
obtain the following capital adequacy ratios:
5
A1 is the first A-graded loan, C2 is the second C-graded loan, and so on.
Note that the capital allocated to the last segment does not cover risk and is in fact the most
risky segment of the three. This means that the last group does not meet the other
conditions.
The table below summarizes loan distribution by segment:
Table 7.4
Grade
Grade
F1
Grade
A1
$
4,728
$
5,528
$
7,728
C2
C4
D1
$
$
$
3,204
4,912
5,320
$
$
$
3,138
4,831
5,042
$
$
$
5,848
5,435
5,765
D3
F1
F4
G2
$
$
$
$
20,239
1,933
2,598
1,090
$
$
$
$
15,411
2,411
358
6,467
$
$
1,800
2,317
Total=
$
44,024
$
43,186
$
$
$
2,652
4,929
6,480
$
42,954
Total=
Total=
It is interesting to note that as
, then
is
necessarily the case for all segments. The following table shows the expected default
probabilities, the Rayleigh coefficient, and the correction for correlation:
Table 7.5
Segment i
1
2
3
pi
0.0774
0.1162
0.1339
Ri(Fi,Mi)
0.0998
0.1741
0.3340
Correction for correlation
0.7790
0.5720
0.2753
Using these values, ratios (7.18) and (7.19) can be verified for all portfolio segments. As
expected, the third group defaults:
(
̅
)
∑
{
}
Now, from (7.19) we obtain individual limits that imply capital adequacy:
Given that the limit for loans in the first group is
, all loans in the group clearly fall
within the limit. In the second group, $15,411 exceeds the limit. Finally, as expected, the third
group is the most problematic as only the four smallest loans respect the limit.
It is interesting to analyze the effects of the correlation in this example. The aforementioned
establishes that the first group can have loans of any size, but the riskiest segment must
include only small loans which, although logical, are clearly not the case.
As for capital adequacy, (7.17) is examined and comparing
with
∑
∑
Note how the result is the same as in example 7.1, which verifies that the portfolio has
enough capital adequacy despite the third segment individually breaching all of the
requirements. It is interesting to note the analytical power of the tool developed here, as
if the exercise is restricted to the use of the general model, as in example 7.1, if
individual segments had not been analyzed, the third segment, which is
clearly a group of risky loans, would have gone unnoticed. Finally, it is also clear that
results depend on the segmentation criteria used, as loans can be classified in such a way
that all segments comply with relevant ratios, in which case risky groups would not be
detected. However, the example clearly indicates how to obtain ex ante an idea of
concentration, in the worst of cases, by trial and error. Further research will surely result in a
more systematic search.§
7.4. INCLUSION OF RECOVERY RATES
It is easy to extend all ratios obtained thus far in order to explicitly take loan recovery rates
into account. This leads to less stringent limits in terms of tolerable concentration for different
segments. There are basically two ways to include recovery rates in the analysis. The first
is to define F directly as “losses given default” (LGD) instead of the pending balance, which
assumes that nothing is recovered if debtors default. This would be more in line with current
practices.6 Thus, if we have an estimate of the LGD vector, it can be used directly in
previously obtained ratios without any change, but must be correctly interpreted.
Including recovery rates explicitly implies making an initial transformation of the original loan
vector. Let “ ” be the diagonal matrix whose elements other than zero are the loss rates due
to default. The losses due to default vector for the portfolio are therefore: P = F and let
V ’= 1TP. To include the correlation, the transformation G = SP is calculated. The original
vector is therefore transformed twice, i.e. G = S F. After calculating, we obtain ratios for
capital adequacy, concentration and individual limits, which show how changes in the loss
given default matrix “ ” impact parameters.
Although the suggested transformation results in a very general model, it is difficult to visualize
the impact of changing recovery rates on credit risk, concentration, and capital adequacy. It
should also be taken into account that recovery rates are in themselves random variables and
so this matrix can change depending on economic and market conditions. Hence the desire to
more easily explore the sensitivity of capital, concentration, and individual limit ratios to
changes in recovery rates. In view of this, consider the alternative of assuming, for example 7,
that the portfolio is segmented so that recovery rates are the same for all loans in the group.
Based on the structure in the section above, let “ri” be the recovery rate of loans that default
in segment “i”, so that loss given default is simply
Pi=(1-ri )Fi
Proceeding as before:
∑
∑
∑(
{
)
(
∑
)
}
This expression clearly shows that any change in recovery rates has a twofold impact. On
the one hand, an increase (decrease) in the recovery rate of any segment in particular,
lowers (increases) the importance of the contribution of the concentration of these segments
(left side of the inequality). Furthermore, their contribution to expected loss also decreases
(increases) the numerator on the right, thus increasing (decreasing) the concentration limit. It
is easy to show that the denominator on the right of the inequality behaves as expected,
6
See document “Credit Risk Modeling” of the Basle Committee on Banking Supervision . April
1999.
7
+
This is what CreditRisk does.
decreasing when the recovery rate increases and vice versa.
Finally, we should point out that segmentation for equal losses in view of default is not
necessary. Although this was convenient in order to show the impact of recovery rates on
capital concentration, limits and requirements, the model can be adapted by transforming the
balances vector and preserving the flexibility of the model in order to segment the portfolio in
any way deemed appropriate for undertaking the corresponding analysis to detect risky
segments.
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