PORTFOLIO MANAGEMENT
A Unified Approach to
Credit Limit Setting
by Jeremy Taylor
T
his article examines the use of economic capital to set limits on
credit exposure, extending the framework outlined in a June 2000
article by the same author, and discusses how to structure house
limits on individual obligors and concentration limits on portfolio segments.
The author finds that, at an aggregated level, stress-testing is more appropriate
than portfolio-level limits. The unifying tool is correlation coefficients, which
are as important as they are difficult to measure or estimate.
T
he discussions over reform of the BIS capital
adequacy guidelines are shedding new light
on the role of economic capital as a tool for
managing credit and other banking risk. While the
importance of capital as a cushion against losses, credit
related or otherwise, has long been understood, the science of risk measurement—coupled with the collection
and management of the required data—only recently
has progressed to the point of contributing to the practice of portfolio risk management for the banking book.
To the extent that an institution has gained understanding and confidence in the output of the credit
portfolio model being used, the resulting capital allocations—by borrower, relationship, and various higher levels of customer aggregation—provide an objective
measure of risk that can be compared across the portfolio, across different types of risk, and across time periods. Most banks have only recently gained this capability and are just beginning to realize its potential. Limit
setting offers a relatively straightforward opportunity to
leverage this tool in an area historically characterized by
somewhat arbitrary assumptions and procedures.
Moving Away from the Traditional Approach
While computing limits based on book capital is
not unusual, it tends to be done crudely as a fixed percentage of capital—enough to accommodate exposure
at the time plus an allowance for growth—and without
attention to the varying risk characteristics of individual
borrowers within the segment. House limits (on individual obligors and on relationships) may have even less
foundation in terms of a tie-in to capital. Ditto for country limits.
By computing a limit in dollars of capital and then
measuring utilization in economic capital terms, a limit
structure can be devised that incorporates and reflects
variability in default, recovery, and drawdown rates, as
well as the all-important correlation characteristics. And
borrowers can be grouped according to the underlying
similarities in their loss patterns. All of this provides a
more flexible, consistent, and defensible structure that
can adjust automatically to changes in the size, composition, and risk profile of a portfolio over time.
Why Are Limits Necessary?
In a world in which credit losses recurred at the
same x% rate each year without variation, there would
© 2002 by RMA. Taylor is VP, Portfolio Risk Management, Union Bank of California. The views expressed in this article are his
and not necessarily those of the bank.
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The RMA Journal July/August 2002
A Unified Approach to
Credit Limit Setting
be no need for capital (forgetting about market and
operating risk); losses would be covered by the expected loss (EL) premium over the cost of funds and
accounted for by the loss-reserving process. But in reality, of course, the x% is only a long-run average rate
around which there’s year-to-year variability. Capital is
required to absorb those unexpected losses when they
exceed their mean value. And when they greatly exceed
it (for example, a severe recession), the question arises
as to whether capital is adequate to ensure the institution’s solvency—and with what level of confidence?
Uncertainty and variability in default rates and
other components of credit loss is a concern for capital
management. But of even greater concern is the covariability between loans, which can be thought of as the
glue in concentrations. In a large portfolio without large
individual exposures, covariances (that is, the off-diagonal terms in the variance-covariance matrix of expected
defaults) greatly outweigh the effect of the variances.
Limits are necessary to guard against large single or correlated credit events and the erosion of capital they produce.
Limits promote granularity by restricting exposure to any single borrower and promote diversification by restricting exposure to any grouping of borrowers.
As the banking industry evolves toward more
active forms of loan portfolio management, limit setting
can be thought of as a front-end, more passive tool—
relative to the buying and selling, securitizing, and
hedging of credit risk. As a way to measure and monitor
concentrations, limits can provide guidance for the
deployment of other, more active tools, such as those
that identify where to build up exposure or where to
sell off or hedge. Grooming the portfolio in this manner
can help nudge it toward the elusive “efficiency frontier” of modern portfolio theory, minimizing portfolio
risk at any given level of expected return.
A Contrary View
In theory, with a sufficiently fine-tuned RAROC
(risk-adjusted return on capital) model and the pricing
discipline to go with it, concentrations would either be
compensated for (by a rising risk premium required to
reach the target RAROC, as concentration risk increases) or cut off (should potential customers refuse to pay
that higher pricing, relative to what other, presumably
less concentrated institutions may be charging).
In practice, however, two factors build a strong case
for having exposure limits:
1. Pricing discipline may wane in the face of competition and the pressure to protect relationships and to
2.
leverage industry success. The result may well be
accumulation of exposure to levels that do pose
concentration risk. As it continues, pricing (which
includes the risk premium built into the lending
rate but also other credit and noncredit revenues in
excess of costs) may not compensate for the incremental risk.
Credit portfolio models may not be fully sensitive to
the effect of build-ups in exposure. While correlations
exert a powerful influence on credit risk capital,
they’re also extremely difficult to estimate, and they
are not always stable. In particular, they tend to climb
in periods of stress, just when they matter most.
In this environment, limits provide an important
backstop to the portfolio-balancing impetus that a
RAROC model alone can provide.
The Guiding Principle of Limit Setting
An economic capital-based approach to credit limit
setting starts with a question: How much of its capital
(or earnings) is an institution willing to put at risk to
any one obligor, industry, country, and so forth? This
approach is consistent with the way that VaR (value-atrisk) models approach the question of market risk and
apply it to the setting of market limits.
Viewed this way, a limit should be expressed in dollars of capital, as some share of available (book) capital,
depending on various risk-related characteristics of a
borrower or group of borrowers. Applying the same principles to calculating use of the limit suggests that exposure amounts be converted into equivalent dollars of
economic capital to reflect the amount of the bank’s capital being put at risk as a result of that loan or group of
loans.
Using economic capital in this way offers several
advantages over the more traditional approach of viewing concentrations as actual dollars of commitments or
outstandings:
• It represents “units of risk.” For example, $1 of
risk grade 3 exposure is allocated less capital and,
therefore, counts less heavily against a segment’s
limit than $1 of risk grade 5 exposure.
• It’s a single measure to capture exposure, including
outstanding, undrawn, and other off-balance-sheet
amounts.
• It’s fully consistent with the RAROC methodology
being deployed by banks for other risk management and performance measurement purposes.
• It’s a measure that can be compared directly with a
bank’s available capital.
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A Unified Approach to
Credit Limit Setting
The Role of Correlations
From a regulatory and a creditor perspective, a
bank holds capital to protect against losses—more precisely, the unexpected losses (UL). The expected losses
are anticipated and absorbed by the EL premium. In
recognition of the uncertainty in EL (that is, the yearto-year variability in actual losses experienced), a risk
premium is charged on top of the EL premium, which
hopefully suffices to provide some targeted RAROC.
The amount of capital allocated against a loan is a
multiple of its UL, defined as one standard deviation of
the loss distribution. That multiple is determined by
the targeted agency rating; a higher rating means a
lower probability of the bank experiencing losses that
push it into insolvency, which, in turn, means holding
more capital. UL reflects two things:
1. Variability of EL components—the default, recovery, and utilization rates.
2. Correlation of EL across borrowers.
Figure 1
Effect of Loss Correlations
Unexpected loss
2
1
0
Rising
correlation:
2 > 1 > 0
Expected loss
2.
Lower covariances: Diversifying to avoid concentration risk.
Note that concentration risk really has two sources.
Systematic risk factors—business cycle swings, for
example—impact all or a large number of borrowers in a
similar way. Granularity effects occur in those portfolios
not fine-grained enough to have diversified away all
idiosyncratic risk. In a typical larger-bank portfolio—
diversified along industry, geographic, and other dimensions and without large individual exposures—risk is
predominantly what’s systematic across borrowers.
As laid out in Figure 2, the full stand-alone risk
(UL) of a borrower can be decomposed into the idiosyncratic (or diversifiable) risk and the systematic risk.
The former, in turn, can be split into the following:
• The risk that has been diversified away (by filling
the portfolio with relatively small, relatively dispersed exposures).
• The risk that has not been diversified but could be.
That still-diversifiable portion, together with the systematic risk, constitutes a borrower’s contributory risk.
Figure 2
The Components of Unexpected Loss
Diversified risk
Diversifiable risk
Systematic risk
Figure 1 illustrates how, for a given level of EL,
higher loss correlations translate into higher UL and,
therefore, higher economic capital allocations. This is
because higher correlations increase the likelihood and
severity of big losses when a large number of related
borrowers get into difficulty concurrently—the opposite
of a correlation-free constant x% loss rate every year.
It’s the periodic spikes in loss experience due to borrower correlations that will produce a higher UL and a
more skewed right tail on the loss distribution. And it’s
because of the possibility of correlated credit events
imposing occasionally magnified losses on the portfolio
that credit models suggest the need for capital at several times UL.
Credit risk can be mitigated (and economic capital
reduced) in two ways:
1. Lower variances: Taking on transactions with lower
inherent (stand-alone) risk.
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The RMA Journal July/August 2002
S = Stand-alone risk
C = Contributory risk
Measuring Correlations
Correlations—the key piece of this exercise—perform double duty. As described earlier, they drive contributory risk and, therefore, the calculation of economic
capital. But they also play a critical role in constructing a
limit system. Starting with an individual obligor and
going all the way to the full loan portfolio, differences in
loss correlation will define and parameterize the groupings of borrowers to which each limit will apply.
As important a role as they play, however, correlations are also the toughest part of credit portfolio modeling. This stems from the unhelpful fact that default
correlations aren’t observable—certainly not at the
obligor level. Logic tells us that the default and loss
probability on a loan to United Airlines is correlated
with the likelihood of Delta defaulting, but we have no
A Unified Approach to
Credit Limit Setting
default history (for either company, let alone both) to go
by. Segment-level default data (for example, how the
average default rate for airline carriers changes in
response to higher average defaults in some other
industry segment or in the economy as a whole) can
provide some indication. But it’s not particularly robust
in terms of the quantity and quality of data available.
Nor is it the whole picture.
Conceptually, the default correlation between two
borrowers, 1 and 2, in industries A and B, respectively,
is given by:
D1,D2 A,B A B
where A and B are the intrasegment default correlation coefficients (for example, the average probability of
any two borrowers in A defaulting together) and A, B is
the intersegment correlation (that is, the relationship
between changes in the average default rate for A vs. B).
Segment-level data will address the latter, but the second two terms on the right side require the companylevel, intrasegment effects—the unobservable piece.
Instead, the modeling and estimation of default
correlations derive from asset correlations, which look at
the co-movement in asset values between firms, or
between one firm and the portfolio as a whole. In the
context of a Merton-type, options-theoretic approach,
fluctuations in firms’ asset value produce default only to
the extent that it dips below the liability-determined
default point. This is why default correlations are small
(mostly in the low to mid single digits, in percentage
terms) relative to asset correlations.
Moving from asset correlations to default correlations requires factoring in the default probabilities. For
two firms, 1 and 2, this means a default correlation,
D1, D2 , as follows:
D1,D2 D1,D2
D1 D2
E(D1,D2) E (D1) E(D2)
D1 D2
E(D1,D2) DP1 DP2
(DP1 (1 DP1)) (DP2 (1 DP2))
where D is default, DP is default probability, and E
(D1, D2) is the joint probability distribution, modeled
from the joint distribution of asset values. For zeroasset correlation, E (D1, D2) = E (D1) E (D2)—that
is, the joint probability is simply the product of the
stand-alone probabilities, like the 25% probability of
getting two heads in two coin tosses, which results in
zero default correlation. As asset correlation rises, and/or
as DP rises, so does default correlation.
But there’s another wrinkle here. Asset correlations
(indeed, asset values) aren’t observable, either. Instead,
the usual starting point is equity correlations, which are
readily obtainable and which serve as a reasonable
proxy for asset correlations but with two important
exceptions: highly leveraged firms, and low credit-quality firms.1 In both cases, relying on equity correlations
will result in asset correlations being understated.
In summary: We need default correlations but can’t
observe them. So we fall back on asset correlations—
but we can’t observe them, either. So we start with
equity correlations, being mindful that they’re an
imperfect proxy for asset correlations. But there’s something even easier.
Sources of Data
A correlation engine is an integral piece of any credit
portfolio/RAROC model. An off-the-shelf model, like
KMV’s or CreditMetrics’, comes with the required capability to process the customer-level data from a bank’s
portfolio. For example, KMV’s GcorrTM (Global
Correlation Model) is a multi-factor model for determining pair-wise asset correlations, for translation into default
correlations using KMV’s expected default frequencies
(EDF). Reliably measured correlations are critical for the
allocation of economic capital—that is, having a measure
that’s sensitive to any buildups in concentration within a
portfolio and to the complex interplay between those
borrowers and the rest of the portfolio.
Correlation data fulfills a second function—guiding
the grouping of borrowers into like segments, based on
principles laid out in the next section. How tightly
glued (that is, how closely correlated) those borrowers
are will be shown to guide the appropriate limit setting
for any such grouping. For this purpose, less precise data
will suffice—getting them correct in relative terms,
with roughly correct differentials between them. For
that, there may be intersegment default correlation
data, drawing on historical bond default rates, for example.2 But again, we quickly run into the void of
intrasegment correlations, the missing half.
For those with access to a bells-and-whistles credit
model like KMV’s Portfolio ManagerTM or
CreditMetricsTM, introduced in 1997 by J.P. Morgan, the
required intra- and intersegment correlations can be
obtained by generating the required subsets of pairwise correlations—that is, against firms in the same
industry (or other segmentation), then against firms not
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A Unified Approach to
Credit Limit Setting
in the same industry. It bears emphasizing that this is
likely to be the only manageable way to access the full
set of data required for all of the pieces of the limit-setting framework outlined in the following sections.
Defining Concentration Segments
Segments (most obviously by industry and geography, though others are possible) should be constructed
according to the loss correlations3 between borrowers.
Within a segment, correlations should be relatively
high, even if something less than perfect—that is, < 1.
Between borrowers in different segments, they should
be relatively low (even if > 0). Carving up the portfolio into segments should not be dictated by how large or
important a group of borrowers is but rather by how
closely correlated they are.4 If there’s too much glue
binding a group of borrowers, then that’s the relevant
grouping for determining correlated loss potential and
how much capital to put at risk lending there. This
principle suggests that some borrowers may fall into
more than one segment (for example, a computer retailer, which presents both technology and retail industry
risk), while others (for whom the systematic risk component is less important) may not fall into any.
House limits apply to borrowers and/or relationships. For example, a borrower is effectively a concentration segment with = 1, while a relationship is one
where < 1 (that is, not necessarily the case that one
borrower in a relationship defaulting means that all the
rest will) but is considerably higher than within a typical industry segment.
Setting Concentration Limits
Let’s start at the segment level, somewhere
between borrowers and relationships at one end of the
spectrum (Figure 3) and the total portfolio at the other.
We’ve constructed our segments as just described. How
do we set limits on their exposure?
First, we allocate a certain percentage of book capital to each segment (we’ll return shortly to look at how
that percentage might be set). That percentage can
then be adjusted according to each segment’s risk-related characteristics—for example, its strategic importance, its short-term prognosis, and, perhaps, its riskadjusted profitability. That gives a limit expressed in
dollars of capital at risk. In order to calculate how much
of its limit a segment is utilizing, we can bring in the
economic capital allocation, which measures how much
actual risk is attributable to the bank’s lending to that
segment.
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The RMA Journal July/August 2002
As we move from borrower to relationship to segment to portfolio, the within-group correlation coefficient will steadily fall, implying lower concentration risk
and, therefore, the ability to take on increased exposure
without putting the bank’s capital at undue risk.
Exposure limits should be scaled accordingly—that is,
the limit should rise with the decline in (the square root
of) . A crude approach, in which all concentration segments (all industry groupings, for example) start with
the same x% of book capital, is certainly simpler to do
and less demanding data-wise. For that approach, intuition and economic logic may suffice for going through
the exercise of grouping borrowers into segments. But
with the benefit of the required segment-level data,
this can be done more flexibly, more sensitively, by giving lower- segments a higher base-capital allocation.
This relationship is depicted in Figure 3.
Figure 3
A Risk Continuum
House limit:
Relationship
House limit:
Obligor
Major segment
limit
Subsegment limit:
Industry, country...
As (intra-group) correlation
Portfolio
limit?
...
... limit amount
Correlation data can allow the base capital allocation for each segment’s limit calculation to be appropriately scaled up from a starting point. Which starting
point? The obligor level (with its = 1), most obviously,
though it could also be a concentration segment, perhaps the one with highest intrasegment correlation. But
how do we anchor that starting point? It could be by
working backwards—allocate a base capital amount that
produces a reasonable base limit, relative to existing
exposure and comfort levels. As a refinement of that,
the institution could look for internal agreement on the
maximum loss that it’s prepared to take on any one borrower or credit event.
House Limits
Suppose a bank’s concentration segments exhibited
an average within-segment default correlation of 4%.
The relationship between house limits (HL) and concentration limits (CL) should then be approximately as
A Unified Approach to
Credit Limit Setting
follows:
HL = 0.2 CL , where 0.2 = 0.04
Some further downward adjustment of that fraction
may be warranted as a granularity adjustment—that is,
to recognize the typical size dispersion of exposures
within a concentration segment (for example, more than
5 [ = 1/0.2] times as many borrowers).
This would express house limits in dollars of capital,
as was done earlier for concentration limits. But it may be
more practical to convert and communicate them in dollars of commitments, as a guide to lenders and line management. If month-end or quarter-end analysis reveals a
segment to be over its concentration limit, the institution
can inform the line of whatever measures then come into
effect (as discussed below). But it will rely on individual
lenders to monitor and comply with house limits as they
go, which is more easily done in commitment dollars
than economic capital dollars.
Converting a limit figure into dollars of commitments requires dividing it by an appropriate economic
capital allocation factor (that is, capital per dollar of
commitments, from the bank’s RAROC model output),
differentiated by major risk criteria, such as risk grade
and line of business. For example: If a limit for a risk
grade 5 borrower in a particular area were $10 million of
economic capital and the capital allocation per dollar of
commitments were $0.05, then the house limit would
be $200 million in commitment terms.
Whereas concentrations can be computed and tested on a regular (at least quarterly) basis, it may make
sense to have house limits set and maintained for longer
periods—say, annually. Among other things, this
reduces the problem of having individual borrowers fall
out of compliance (that is, go over house limit) without
a change in their commitment amount.
Relationship Limits
Relationship limits are similar to house limits, but
they are aggregated across obligors to the relationship
level. The need for relationship limits in addition to
house limits is an open question. Furthermore, there
are (even bigger) data issues that come into play—for
example, how do we measure the likelihood of all borrowers within a relationship going into default together?
Intuitively, the number is high, certainly higher than for
any concentration segment, though how high will
depend importantly on an institution’s aggregation rules
for defining what constitutes a relationship. This is not
easily quantified; neither is the underlying correlation
within a relationship likely to be constant across a bank.
One important example: For commercial real estate,
relationship (that is, sponsor) limits may be higher than
for general commercial relationships. This is because
the dispersion of individual projects with different tenants and more autonomous legal structures means less
likelihood of one defaulting project triggering others
within the same relationship.
The upshot: If separate relationship and obligor
limits are deemed necessary, an assumption about the
degree of commonality within a typical commercial relationship is required to scale up the relationship limit
from the obligor limit. Commercial real estate appears
different enough to warrant separate treatment (that is,
a separate scaling factor).
Country Limits
The theoretical framework outlined here could certainly be applied mutatis mutandi to setting and monitoring country limits. Indeed, the earlier discussion of concentration limits was couched in terms that are general
enough to apply to any portfolio segmentation, including a geographic one like countries or regions. In practice, though, there may well be data limitations that
make RAROC determination more complicated and
less reliable for international customers. Just as important, the exercise of setting country limit amounts is
likely to be overwhelmed by the subjective factors—in
particular, the strategic or comfort factors. They’re
intended to capture the institution’s commitment to,
experience in, and comfort with a given country. These
considerations are likely to vary significantly from one
country to the next, much more so than the industry-toindustry differences that a bank’s senior management
might identify. Meanwhile, the data required to draw
comparisons of intra-country correlations is likely to
inhibit that part of the exercise.
Portfolio Limits versus Portfolio Stress-Testing
Continuing in the same direction (that is, to the right
end of the Figure 3 continuum), it’s possible to go to the
extreme of setting a portfolio-level limit on the total lending book, tied to the overall level of capital in the same
way that the various other limits were. The fact that correlation across the whole portfolio would be that much
lower than in any single segment (which a full credit
model would substantiate and quantify) means a correspondingly higher limit than for any segment within it.
However, there are practical problems here, too. In
particular: Whose limit is it? If a portfolio-level limit
were calculated and the bank were getting close to it, it
article continues on page 72
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A Unified Approach to Credit Limit Setting
continued from page 61
becomes an unwieldy management tool. Decisions
regarding overall portfolio size, growth, and composition
are part of board-level strategy, not the short-term
grooming that limit-setting produces.
That said, however, the exercise of comparing the
institution’s total capital to the credit and other risks
(expressed in economic capital terms) that lurk in its
balance sheet and operations is an important one.
Rather than approaching it from the viewpoint of setting aggregate limits on the institution’s assets based on
its available capital, the exercise can be turned around
to one of stress-testing—taking the existing portfolio
and the stresses to which it might reasonably be subjected and assessing the adequacy of bank capital to
absorb the resulting losses.5
Stress-testing is familiar enough at the obligor level
as an assessment of the firm’s ability to continue to
service debt under stress conditions. At the segment
level, it aims to determine relative vulnerabilities across
the portfolio and the stress-scenario impact on loss
reserves. Our concern here is the portfolio (or macro)
level.
In setting concentration limits, the portfolio was
segmented into buckets of relatively homogeneous risk.
Concentration risk stems from overexposure to shocks
emanating from and largely confined to individual
industries or regions. Stress-testing, on the other hand,
recognizes that, in addition to those segment-specific
shocks, there are systematic risk factors (like interest
rate changes, recessions, equity shocks) that affect all
segments, though to varying degrees. These are why ␳
> 0 between segments. Portfolio-level stress-testing is a
substitute for having limits in place to protect against
those macro-level shocks. It looks at the incremental
losses and the resulting erosion of bank capital (defined
to include loss reserves) under stress conditions
deemed “reasonably possible” by historical norms.
Limit Compliance and Breach Management
Crafting a new limit structure designed to signal
unhealthy buildups in concentration won’t do much
good without some teeth. On the spectrum from soft to
hard limits, shown in Figure 4, there are probably benefits to seeking a middle strategy somewhere between
“raising the limit” and an “absolute ban on new business.” The former undermines the integrity of the
process, while the latter places too much faith in it;
there’s a need for some flexibility, given the data limitations already discussed, the drawbacks of the SIC codes
for classifying industries, and other such “real world”
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The RMA Journal July / August 2002
intrusions. Response to the breach of a segment’s limit
may therefore include a tightening of underwriting
guidelines and/or credit approval procedures or perhaps
temporary approval of the over-limit amount until it can
be sold down or hedged.
Figure 4
Limit Compliance
Alternative Breach Management Strategies
SOFT
LIMIT
Liquidity-based
constraints (only
sectoral review: loans that are
saleable can be
•install higher
booked)...perhaps
pricing hurdle
with a temporal
•revisit limit-setting assumptions dimension (i.e., a
•revisit underwrit- temporarily higher
limit, where liqing guidelines
uidity justifies)
Raise Use breach as
limit opportunity for
Absolute
ban on
new
business
HARD
LIMIT
Conclusion
The development and acceptance of economic capital as a risk measurement tool are just beginning to
impact bank portfolio management practices. Credit
limit-setting is an area in which traditional approaches
have tended not to give due attention to the key differences between borrowers or segments in the underlying
risk drivers. By incorporating the variances and covariances in loss patterns, economic capital provides a comprehensive measure of risk, one that can be directly
compared with an appropriate calculation of the amount
of capital an institution is willing to put at risk in lending to a particular borrower or group of borrowers. This
can make the limit-setting process more defensible and
more disciplined, so as to steer the institution away
from the concentration risk that a more focused and
specialized approach to lending can cause. ❐
Contact Taylor by e-mail at [email protected]
Notes
1 For a good discussion of this, see "Measuring Credit Correlations: Equity
Correlations Are Not Enough," B. Zeng and J. Zhang, unpublished KMV
paper, January 2002.
2 For example, Michael Niemira of Bank of Tokyo-Mitsubishi in New York has
done this analysis on high-yield bond defaults, using Fitch data.
3 We'll assume independence in recovery and drawdown rates and focus on
default (vs. loss) correlations.
4 Of course, a larger group of borrowers (e.g., multi-industry, or multi-region)
is likely to be more heterogeneous (i.e., to have lower intra-group correlation).
5 See "Stress-Testing a Commercial Loan Portfolio," J. Taylor, Commercial
Lending Review, March 2002.