systemic risk and heavy tails: the case of banks’ loan portfolio
TRANSCRIPT
Heavy Tails in Bank Loans: An Overlooked Systemic Risk Component
By Sumit Agarwal and Hamid Mohtadi ♣
March 2011
Abstract
We estimate a Pareto distribution for loan losses, as an alternative to the
commonly used Vasicek distribution, using simulated data. A key assumption in the
construction of Vasicek distribution is that firm-level risk is idiosyncratic. It also assumes
that firm exposure to systemic risk is constant across firms (captured by R1/2
). But
profitability may decline differently across firms in a recession in ways that are both
systematic and related to firm characteristics, such as size, sector, or firm-level fixed
effects. Finally, when calculating expected loss under the Vasicek framework, the
probability of default (PD) and loss given default (LGD) of loans are assumed to be
uncorrelated. Yet, evidence points to correlations between PD and LGD especially when
systemic risk is dominant. (c.f. Miu and Ozdemir, 2006, Altman, Reti and Sironi, 2005).
Its possibility has also been discussed in the BASEL final rule (Federal Register 2007 p.
69309).
We use a Pareto distribution to overcome these restrictions. We find a higher probability
of large losses than has been understood thus far. We examine our findings, based on
simulated data that reproduce a bank’s mean PD and LGD values. The findings do show
that the Pareto distribution predicts a higher likelihood of tail events and fits the data
more closely, when compared with the Vasicek distribution. The threshold or cross-over
point at which this “tail” result obtains corresponds to PDs ranging from 1% to 2%, a
range found in the mid to mid-high risk segments for many banks’ portfolios. The use of
Pareto distribution is consistent with the new thinking about the role of Power Law in
explaining tail events (Gabaix, et. al, 2006, 2008).
Keywords: Systemic Risk, Heavy Tails, Capital Adequacy, Bank Regulation, Financial
Institutions
JEL Finance Classifications: G2, G3
♣ Agarwal is Senior Finanical Economist, Federal Reserve Bank of Chicago, [email protected],
Mohtadi (contact author) is Professor of Economics, University of Wisconsin at Milwaukee,
[email protected] and [email protected]. The views expressed here do not represent those of the Federal
Reserve Bank of Chicago or the Federal Reserve System.
1
1. Introduction
Over the past year there have been a number of influential research papers that
had analyzed the causes, consequences and remedies of the 2008 financial crisis (Acharya
and Richardson (2009); Acharya, et. al (2010); Brunnermeier, et. al. (2009)). These papers
argue that the risk models failed to adequately prepare the marketplace for the collapse of
the market for mortgage-backed securities and credit derivatives. Furthermore, the faulty
assumptions underlying these models blindsided otherwise vigilant market participants
about the risks and brought about the global financial system. It is widely believed that at
times of crisis asset-price volatility does not follow any statistically useful probability
distribution . Hence, in this paper, using simulated data we estimate a Pareto distribution
for loan losses for banks as an alternative to the commonly used Vasicek distribution.
The advantage of using a Pareto distribution is that it is free of the assumptions and
restrictions underlying the Vasicek distribution, but has similar Heavy-tailed properties.
Using this distribution, we find a higher probability of large losses than has been
understood thus far. Our finding is supported both analytically and by our data.
Historically, systemic financial events, such as banking panics, date back to at
least the late 1800s in the US. Yet, attempts to clarify, define and quantify systemic risk
are relatively new.1 Conceptually, answers to the question of what constitutes systemic
risk vary widely and range from contagion effects to shocks that cause widespread
bankruptcies (Schwarcz, 2008). Operationally, however, in such areas as in banking and
the implementation of Risk Based Capital, the treatment of systemic risk has become
relatively standardized. This is due, for the most part, to the well known model of loan
loss distribution, originally developed by Vasicek (2002). Vasicek considers how the
1 See for example, de Bandt and Hartmann (2000) for a detailed and informative survey.
2
presence of systemic risk would affect a bank’s loan assets. The result is a Heavy tailed
distribution. Vasicek’s formulation proceeds from the assumption that correlated assets
can be decomposed into two additive, separable and independent components; a systemic
risk factor (source of asset correlation) and a firm-level idiosyncratic risk factor
orthogonal to the systemic risk factor. The resulting distribution of the losses is a well
known Heavy-tailed distribution that has been incorporated into the BASEL economic
capital formulation.
A key assumption that goes into the construction of Vasicek distribution is that
systemic risk and firm level risks are additively separable and do not interact with one
another, as firm level risks are assumed to be purely idiosyncratic. However, if this
assumption does not hold (see below), Vasicek distribution is not the correct distribution
to model systemic risk events. In this paper, we focus on an alternative Pareto
distribution and show, using simulation data, that this distribution (a) is better supported
by the data for near-tail to tail events, compared with the Vasicek distribution, and (b)
predicts a higher likelihood of tail events than Vasicek distribution.
Another contribution of our paper is to account for possible correlation between
probability of default (PD) and loss given default (LGD), when calculating expected loss.
Recently, there have been concerns about the accuracy and adequacy of the Vasicek-
BASEL formulation due to observed correlations between PD and LGD. This is because
Vasicek’s distribution predicts PD while LGD and EAD (exposure at default) are
assumed constant. Yet, the existence of PD-LGD correlation has been acknowledged in
3
the literature (c.f. Miu and Ozdemir, 2006, Altman, Reti and Sironi, 2005). Its possibility
has also been discussed in the BASEL final rule (Federal Register 2007 p. 69309). 2
Accordingly, we approach our task in two parts. Following a brief presentation of
our general modeling framework (Section 2), we examine the analytical underpinnings of
BASEL-Vasicek distribution, and evaluate the alternative Pareto distribution, comparing
our findings with those from Vasicek’s (Section 3). However, this is a controlled
experiment and produces only a partial picture. The full picture emerges when we add
PD-LGD correlation to the mix so that the distinction between the distribution of the
actual loss and the distribution of PD becomes non-trivial. This task belongs to the
second part (Section 4). Our conclusion is presented in Section 5.
2. Basic Framework
Let iA be the value of the ith borrower’s assets. The random component of this
asset iX is typically assumed to be normally distributed, consisting of an economy-wide
component, Y, to represent systemic risk and an orthogonal firm-level component, Zi, to
represent idiosyncratic risk and is standard normal, Zi ∼N(0,1). This is shown by the
asymptotic single risk factor (ASRF) approach which is a general framework for
determining regulatory capital requirements for credit risk, as follows:
ii ZRYRX2/12/1 )1(. −+= , (1)
where R represents correlation between asset iA and the single common (systemic factor),
Y, such that R1/2
Y reflects the “company’s exposure to the common factor” (Vasicek,
2 Yet the calculation of economic capital in the final rule does not take explicit account of PD-LGD
correlation, arguing instead for the use of downturn default as a way to ensure caution.
4
2002). Notice that company exposure here is assumed to be constant across all firms and
quite independent of firm level risk. The latter is assumed to be entirely idiosyncratic.
However, the simplified formulation in (1) may be too restrictive for several
reasons. For example it identifies firm-level risk only with the idiosyncratic factor, or it
assumes that firm exposure to systemic risk is constant across firms (captured by R1/2
).
On the latter point, for example, profitability could decline differently across firms in a
recession in ways that may be related to firm characteristics, such as size, sector, or firm-
level fixed effects. Finally, assuming a linear functional form as in (1) may limit its
applicability.
If equation (1) is too restrictive, the Vasicek distribution will not hold. To
maintain the Heavy-tail structure of the Vasicek distribution without the accompanying
restrictive assumptions that lead to its derivation, various alternative forms of Power Law
family of distributions present themselves, since these distributions have less restrictive
forms, have very simple structures, and still share the Heavy-tailed property with
Vasicek. In this paper we focus on Pareto forms in this family of distributions, for
reasons discussed later, and compare our results with those found from Vasicek.
A second (distinct) assumption of the Vasicek formulation is the assumption of
the independence of PD and LGD. With an emphasis on the distribution of PD, this
implies,
)(..)( PDVLGDEADLE = (2)
where V(.) is the Vasicek density function of PD consistent with (1). A more general
formulation of (2), however, would allow for potential interdependencies between LGD
and PD, as was discussed in the introduction. This point will be explored later in Section
5
4. The issue of PD-LGD correlation may be in fact related to the issue that firm exposure
to systemic risk may not be constant across firms (first point). This may be especially true
in downturn conditions: The difficulty for a bank to recover its losses from an obligor
(and thus the size of its LGD from that obligor) is more sever in a downturn as a the
obligor revenues as well as the value of obligor assets are less during a downturn, where
the probability of default is also known to be higher.
To keep the treatment of systemic risk distinct from the issue of PD-LGD
correlation, we conduct a controlled experiment. First, we examine the implication of
relaxing the non-linearity assumption only, comparing our findings with Vasicek’s. In the
second, we add the issue of PD-LGD correlation and re-examine our findings.
3. An Alternative Formulation: The Basic Model
Power Law distributions, β−= xxP )( , are simple and can be viewed as easy-to-
use alternatives to Vasicek distributions in ways that are free from the restrictive
assumptions discussed above. Power Law finds application in a variety of instances in
which the underling process is a statistically skewed. In financial markets, Gabaix, et. al.
(2006) find that the process underlying the distributions of the volume and returns follow
Power Laws for large trades and explain that by the existence of large “market makers”.
The key discovery in physics, known as Scale Invariance, has allowed both economists
and physicists to be able to generalize the presence of Power Law in numerous physical
and financial phenomenon. Newman (2005) describes many such instances, ranging from
word frequencies, to web hits, to magnitudes of earthquakes, and the intensities of wars.
Spagat and Johnson and Spagat (2005) show Power Law at work in describing the
number of attacked in a war, applying their analysis to US war in Iraq. Mohtadi and
6
Murshid (2009a, 2009b) show that a form of Power Law, in the form of extreme value
distributions describes the instances of terrorism attacks. Thus the present perspective on
the examination of power law in loan loss distribution, while new to this area, follows a
rich background of analysis and examination by physicists and economists.
In this paper, we focus on a specific form of Power Law distributions, i.e., the
Pareto distribution, as follows:
α−>> =≥ )()|( 0
m
xxx
xxXP
m (3)
Where X is a random variable, α is an unknown parameter, and mx is the threshold value
to be defined from the data. The density function for this distribution is given by,
)1(..)(
+−= ααα xxxP mX (4)
The choice of Pareto is motivated by its relation to another statistical
phenomenon, representing rare events, known as Extreme Value Distribution: It turns
out that when the threshold mx in equation (4) and, by extension, the value of x, is large
(i.e., focus on the tail of the distribution), the Pareto distribution is a special case of the
Generalized Pareto family, with the following density function:
)1
1()(
11
)(ξ
σ
µξ
σ
+−
��
���
� −+=
xxf (5)
where µ and σ are location and scale parameters, and ξ characterizes thickness of the
tail. In turn, the Generalized Pareto distribution is related to the family of Generalized
7
Extreme Value distributions (GEV), 01 1
1
≥��
���
� −+
��
�
��
��
��
���
���
���
� −+−=
−
σ
µξ
σ
µξ
ξx
;x
exp)x(G .
3 (see Coles, P. 75). In fact Gnedenko (1943) has shown that,
)(.)(lim
/1 xfxxGx ∞→
−→ ξ (6)
where f is some slowly varying function. As such, the Pareto parameter, �, and the GEV
tail parameter ξ have approximately inversely related:
)1/(1 αξ +≅ for large x. (7)
It would therefore suffice, as far as the tail of the distribution is concerned, to use Pareto
distribution—which has fewer parameters and is easier to approximate—as a reasonable
representation of the family of extreme value distributions.
3.1 Data Construction for the Basic Approach
As mentioned, in the basic model we ignore any PD-LGD correlation, focusing instead
on the systemic risk issue at large.
For the purpose of comparability it is useful to scale our random variable PD
examine our Pareto distribution in terms of P(loss) instead of P(PD). The two
distributions are as follows:
]2/))((/2/))()(*1(exp[1
);( 21211PDNRPDNPDNR
R
RPDPDV
−−− +−−−−
=
(8)
Where PD is an average PD over the loan population, and
3 Specifically, 0<ξ corresponds to the Weibull family of distributions (bounded tails), 0=ξ
corresponds to the Gumbel family (light tails) and 0>ξ corresponds to the Fréchet family of distributions
(heavy tails).
8
)1()0|(
αα +−=>>> ����� mm aLP , (9)
where L is a random variable representing absolute loss and m� is the threshold value of
L. We now need to make the two measures of PD and loss commensurate. Since LGD
and EAD are constant, in this section, the size of loss is strictly proportional to PD. By
normalizing the two distributions, therefore, the question of “scale-dependence” in this
transformation of the variables is resolved.4
We choose the sequence of 1000 PDs while allowing PD to vary from 0.025 to 1
with steps of 0.005, such that mean PD, i.e. PD =1.5% which is close to a typical bank
average portfolio PD. For this section, we keep LGD=1, and EAD=10,000. The
threshold value of loss, i.e., m� must be chosen. To obtain this threshold value, we first
note the linearity of E[L] in m� and use this to construct the mean residual life plot,
m
1
/ ���
�
−�=
m
m
N
N
k
k from the set of points mk �� > against different values m� and search for
the linear portion of the plot. This corresponds to the appropriate range of data for a fit by
Pareto distribution. This is shown in Figure 1 before the value of L is transformed to the
range [0,1). As can be seen, the linear behavior occurs in the range of 100 to 800 (losses
are in thousand dollars). We thus choose a threshold value of 100 in this linear range to
maximize the total number of records in our sample.
4 While the Pareto density function is automatically normalized over its entire theoretical range [xm,, ∞),
manual normalization is needed over the actual range of simulated data. Moreover the Vasicek pdf does
need to be normalized: Thus we have,
�+−
+−
=max )1(
)1(
�
�
���
��
m
da
aP
m
m
nαα
ααand
�=
max
)(
)()(
PD
PD
n
m
dPDPDV
PDVPDV where PDm
is the value of PD corresponding to m� in Pareto distribution.
9
Figure 1. Finding the Threshold for Pareto Distribution
The Maximum Likelihood estimation of the parameter α of the Pareto
distribution can be derived analytically.5 Using our simulated data we find, α = 0.63 ±
0.02. Casting this is terms of the parameter of the Power Law distribution, i.e.,
β−≈>> ���� )|( mLP , this value corresponds to �=1.63 ± 0.02. Some comparisons of
this value with other findings are worth noting. In the equity market, Gabaix, et.al. (2007)
finds a coefficient of � close to this number (1.5) to for the distribution of aggregate
trading volume. In the physical world, Newman (2005) cites a range of values for �
ranging from 1.8 (intensity of wars) to 3.14 (diameter of moon craters). Still, a common
value of � is often found to be in the range of 2.5 to 3. For example, Gabaix et. al. (2007)
report a value of �=3 for the distribution of returns to stock returns. We will revisit the
5 The likelihood function is given by )log().1()log(..)log(.)( �+−+= im llNNL αααα .
Maximization ( 0/)( =αα ddL ) leads to )]log(.)log(/[ mi lNlN −= �α , with the error term to be found by
N/ασ = (See Newman, 2005).
10
comparison between Pareto and Power Law distribution once again in Section 4, where
we re-estimate � for full simulation with PD and LGD both varying.
We now estimate the competing Vasicek distribution. For this distribution, we use
R=0.12 and R=0.25 for the parameter values of asset correlation R.6 Results are then
depicted in Figures 2 and 3 focusing only on the tail of the two distributions.
Figure 2: Vasicek-Pareto Tail Comparison: PD Variations only, R=.12
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
0.0
1
0.0
2
0.0
3
0.0
4
0.0
5
0.0
6
0.0
7
0.0
8
0.0
9
0.1
0.1
1
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2
0.1
3
0.1
4
0.1
5
0.1
6
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7
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8
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9
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1
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2
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3
0.2
4
0.2
5
0.2
6
0.2
7
0.2
8
0.2
9pareto
Vasicek
6 For wholesale portfolios Basel prescribes the formula, PDxeR .5012.012.0 −+= (cf., Federal Register, 2007
P. 69411) where PD is the PD for PD rates. For our portfolio with PD =.015, this formula implies that
18.≅R . Our chosen values of R of 0.12 and 0.25 encompass this value, but also provide sufficient
variation to allow us to study the impact of R on the Pareto-Vasicek comparison.
11
Figure 3: Vasicek-Pareto Tail Comparison: PD Variations only, R=.25
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
0.0
1
0.0
2
0.0
3
0.0
4
0.0
5
0.0
6
0.0
7
0.0
8
0.0
9
0.1
0.1
1
0.1
2
0.1
3
0.1
4
0.1
5
0.1
6
0.1
7
0.1
8
0.1
9
0.2
0.2
1
0.2
2
0.2
3
0.2
4
0.2
5
0.2
6
0.2
7
0.2
8
0.2
9
pareto
Vasicek
While the two distributions exhibit somewhat similar tail behavior, there is a point at
which the Pareto distribution crosses over Vasicek from below, and stays above the latter
from that point on. This crossover point occurs at the PD =4.3% for figure 2 and PD=7%
for figure 3. In short, the Pareto distribution predicts a larger probability of observing
default among riskier loans than the Vasicek distribution does. It is noteworthy that the
cross-over PD value of 4-6% is at still far less than the worse rating grade in the S&P
scale of CCC of 25% for CCC, suggesting that its is in the plausible, PD range. This is a
remarkable finding.
The slight increase in the crossover point for a higher asset correlation value
means that the Vasicek distribution shifts somewhat to the right, reducing the range of
underprediction relative to Pareto. At the first glance, this seems to suggest that a higher
12
R would cause the tail of the Vasicek distribution to behave more like Pareto, moving
towards convergence with the latter. However, this appearance may be faulty. For, if
firm-specific exposure to systemic risk exists, as per discussion in Section 2, following
equation (1), then it follows that 0),( ≠iZYE . Now consider variance of Xi:
),(.)1(21)( ii ZYERRXVar −+= (10)
In this relation, since 0),( ≠iZYE , then Var(Xi) departs from unity, which is the basis of
the Vasicek’s standard normal distribution, depending on how large is the term
)1( RR − . Evaluating this term for our two values of R=0.12 and 0.25, we get,
)1( RR − =0.32 and 0.43 respectively. Thus, the larger is R (up to a maximum of
R=0.50) the greater does Vasicek distribution depart from the underlying assumption
upon which it is based.
While the above argument provides a theoretical critique of the Vasicek
distribution for higher values of R, the key question is which distribution ultimately fits
the actual data better. To answer this question, we have to move to the full model where
total loss, rather than PD alone is considered. Here any possible PD-LGD correlations are
also taken into account. This is examined in the next section.
4. The Full Model
In this section, we consider the full portfolio loss rather than PD. The two would produce
identical distributions, if LGD is independent of PD. But if that is not the case, as some
recent evidence suggests it may not be (see the Introduction), we must add PD-LGD
correlation to the mix. Thus, we study Pareto distribution of the entire loss variable, per
13
equation 9 without the limiting assumption of the previous section in which LGD and PD
were held constant. This approach, however, poses a challenge to our comparison of the
Pareto distribution with the Vasicek distribution. This is because the Vasicek distribution
is presented in the literature (cf. Vasicek 2002 and BASEL II “Final Rule”, Federal
Register, 2007) in terms of the PD only, always holding LGD and EAD constant. In
relaxing this assumption we are mindful of the fact that the support in the original V(PD)
is the interval [0,1], whereas the support for V(L) is in principle [0, ∞ ). In practice since
one deals with a finite portfolio, we address this challenge by normalizing the variable L
by dividing it into maxL . This defines a new transformed variable which we will call, �.
i.e., max/ LL=Λ and mean Λ . Then, in analogy to equations 8 and 9 we have:
]2/))((/2/))()(*1(exp[1
);( 21211 Λ+Λ−Λ−−−
=Λ −−−NRNNR
R
RpV
(8’)
and
)1()0|(
ααλλλλλ +−=>>>Λ mm aP (9’)
Where, � is also the transformed value of � .
The presence of PD-LGD correlation means that expected loss will now contain
additional term(s). To see this consider the loss in a portfolio of n loans in its most
general form:
�= iii UIAL (11)
Where, Ai is size of loan i, Ii is an indicator variable (0,1) for default/non-default status
and Ui is the proportion of loan i lost upon default (0�Ui�1). Expected loss then is,
�= )()( iii UIAELE (12)
14
Even assuming that the asset size Ai is independent of its default status and its loss given
default ratio, the covariance between the latter two variables will still remain. Thus,
equation (12) becomes:
[ ]
.).,(..
).|()().().()(
� ��
+
=+=
iiiiii
iiiiii
APDLGDCovPDLGDEAD
AIUCovIEUEAELE (13)
It is readily seen that E(L) reduces to the commonly used expression used in Vasicek and
BASEL formulations, only if PD-LGD covariance is negligible, otherwise one must
estimate E(L) loss as in (13).
4.1 Data simulation for the Full Model
We choose from 1000 random loans as before, but with some differences. First,
we randomly choose 1000 PD values in the interval (0,1). We then randomly choose the
corresponding EAD values in such a way that the weighted average PD of the portfolio
(weighted by EAD), is PD = 1.5% which is the same as for the simple model. Finally,
we randomly choose LGD values subject to a pre-defined correlation between them. We
choose two rather different values of Cor(PD,LGD)=0.1 and =0.5, corresponding to a
low and a high degree of correlation. The covariance term associated with this correlation
(per equation 13) is used only for the Pareto distribution, as the Vasicek distribution,
against which we will benchmark our findings, does not consider this possibility. As a
result of this difference, the X axis must be made commensurate for between the two
distributions. To do this, we resort to the same normalization approach that was adopted
before (i.e., divide by the Max E(L)).
For the Vasicek distribution, we must also choose the asset correlation R. We
choose the same values of R=.12 and 0.25 as in do this, as in the simple model. Finally,
15
for the Pareto distribution, the threshold of 100 is chosen, associated with the linear
region as in Figure 1. The Maximum Likelihood Values of the Pareto parameter and the
corresponding value of Power Law parameters were found to be:
PD-LGD correlation =0.10 Pareto =α̂ 1.60±0.06 Power Law =β̂ 2.60±0.06
PD-LGD correlation =0.50 Pareto =α̂ 1.95±0.07 Power Law =β̂ 1.95±0.07
Figures 4-7depict a comparison of the tails of the two distributions for different
values of R and PD-LGD correlations, while figures 4a-7a depict the two distributions
along with the underlying data against which they were approximated in order to see
which distribution is empirically supported in the region of interest.
Figure 4: Vasicek-Pareto Tail Comparison:
Full Loss Distribution, R=0.12, PD-LGD Correlation=0.10
0.0000
0.1000
0.2000
0.3000
0.4000
0.5000
0.6000
0.7000
0.8000
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
Pa reto
vas icek
16
Figure 5: Vasicek-Pareto Tail Comparison:
Full Loss Distribution, R=0.12, PD-LGD Correlation=0.50
0.0000
0.0500
0.1000
0.1500
0.2000
0.2500
0.1
50.2
0.2
50.3
0.3
50.4
0.4
50.5
0.5
50.6
0.6
50.7
0.7
50.8
0.8
50.9
0.9
5
0.9
999
Pareto
Vasicek
In figure 5 the cross-over point occurs at a loss ratio of 16.25% for PD-LGD correlation
of 10%. Mapping this to its corresponding PD value from our underlying data, we find a
cross-over PD value of 1.1%. In figure 5, we increase the PD-LGD correlation to 50%.
Interestingly this leaves the cross over-point of loss ratio at 16.25%, but does raise the
mapped PD value from 1.1% above to 1.4%. (See below for an explanation.) To put these
numbers in perspective, Moody’s cooperate default rates range from 0 to 13%, as shown
in Table 1.
Table 1. Moody’s Corporate default rates 2002-2007
Moody’s
17
2002-2007
Aaa to Aa3 0.00%
A1 to A3 0.06%
Baa1 to Baa2 0.19%
Baa3 to Ba1 0.39%
Ba2 0.42%
Ba3 to B1 0.53%
B2 1.71%
B3 3.39%
Caa-C 13.725%
Thus, for either PD-LGD correlation values, sub-portfolios in the range of B2 to B3 will
experience a higher probability of loss under the Pareto than under the Vasicek
distribution. Notice that these values, while containing higher risk loans, are certainly
within the “reasonable” range for a typical bank’s sub-portfolio of higher risk loans and
in any case, far less than the 14% PD value found in Figure 2, for the simple model, or
the 13.725 value found in Table 1, above.
The intuition behind the shift to a higher cross-over point in PD, but not in loss
ratio, is probably that a higher PD-LGD correlation at low PD values also implies a lower
LGD value. As a result, one must move to a higher PD value to obtain the same loss
ratio. But it remains interesting that cross over loss ratio does remain the same for both
PD-LGD values. To further examine this and other aspects of our findings we will later
repeat this experiment with the assets correlation value of R=0.25.
But before we do that, we now examine the two distributions against the actual
empirical distributions from simulation data to see which distribution is better suited to
the data. The results are presented in Figure 4a for PD-LGD correlation of 0.10 and
Figure 5a for PD-LGD correlation of 0.50 (in both cases R remains at 0.12):
18
Figure 4a: Vasicek vs. Pareto vs. Data: R=0.12, PD-LGD Correlation=0.10
0.00000000
0.05000000
0.10000000
0.15000000
0.20000000
0.25000000
0.30000000
0.1
0.1
50.2
0.2
50.3
0.3
50.4
0.4
50.5
0.5
50.6
0.6
50.7
0.7
50.8
0.8
50.9
0.9
5
Pareto
vasicek
data
Interestingly the data line has the same crossover point as the two distributions (loss
ratio=16.25%, or PD=1.1%). As the figure shows, beyond this point, the higher tail
probability exhibited by Pareto over Vasicek is also supported by data.
We now repeat the above experiment for the PD-LGD correlation =0.50. This is
shown in Figure 5a.
Figure 5a: Vasicek vs. Pareto vs. Data: R=0.12, PD-LGD Correlation=0.50
19
0.00000000
0.00500000
0.01000000
0.01500000
0.02000000
0.02500000
0.03000000
0.2
0.2
50.3
0.3
50.4
0.4
50.5
0.5
50.6
0.6
50.7
0.7
50.8
0.8
50.9
0.9
5
0.9
999
Pareto
vasicek
data
Similar to Figure 4a, the data support the Pareto distribution’s findings for larger loss
ratios. Notice, however, that unlike Figure 4a, the data intersection point is not for the
same intersection point of the two distributions. In this case the data line crosses over the
Pareto distribution at about L=0.42 which corresponds to a PD ≅ 1.5%. Again, according
to Table 1, this falls on the corporate risk levels between B2 and B3. Notice that the
Pareto line beyond this cross-over point fits the data quite well.
We now conduct our final experiment by raising the size of asset correlation, R.
Figures 6 and 7 show the two distributions for the same two values of PD-LGD as before
but for the asset correlation value of R=0.25.
20
Figure 6: Vasicek-Pareto Tail Comparison:
Full Loss Distribution, R=0.25, PD-LGD Correlation=0.10
0.0000
0.0200
0.0400
0.0600
0.0800
0.1000
0.1200
0.1400
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Pareto
vasicek
Figure 7: Vasicek-Pareto Tail Comparison:
Full Loss Distribution, R=0.25, PD-LGD Correlation=0.50
0.0000
0.0200
0.0400
0.0600
0.0800
0.1000
0.1200
0.1400
0.2
0.2
50.3
0.3
50.4
0.4
50.5
0.5
50.6
0.6
50.7
0.7
50.8
0.8
50.9
0.9
5
0.9
999
Pareto
Vasicek
21
In figures 6 and 7 the cross-over points are at loss ratios of 0.32 and 0.40. These map to
our data for corresponding to PD values of 1% and 2% respectively. These results are
summarized in Table 2, second column and compared to those from Figures 4 and 5,
summarized in the first column.
Table 2: Pareto-Vasicek Crossover points for different R and PD-LGD correlations
R=.12 R=.25
Correlation (PD-
LGD)=.10
Cross-over loss=16.25%
Cross-over PD=1.1%
Cross-over loss=32%
Cross-over PD=1%
Correlation (PD-
LGD)=.50
Cross-over loss=16.25%
Cross-over PD=1.4%
Cross-over loss=40%
Cross-over PD=2%
Comparing this result with Figures 4 and 5, we make two observations. First, we can
readily see that for a given value of PD-LGD correlation, a larger value of R implies a
higher cross-over point. Thus a higher R has the same impact as a higher value of PD-
LGD correlation, but the underlying cause here to do with the behavior of the Vasicek
distribution for larger R. Second, unlike the previous case, where the cross-over loss
ratio remained the same as we moved from a lower to a higher PD-LGD correlation, here
the cross-over loss ratio increases from one figure to the next. Thus, it appears that there
is an interaction effect at work in which the role of PD-LGD in the analysis, itself
depends on the value of asset correlation. In particular, high R values exacerbate the
difference between a low and a high PD-LGD correlation value.
22
Finally, we shall examine the accuracy of our measure with actual data for the
case of R=0.25. This is given in Figures 6a and 7a.
Figure 6a: Vasicek vs. Pareto vs. Data: R=0.25, PD-LGD Correlation=0.10
0.00000000
0.02000000
0.04000000
0.06000000
0.08000000
0.10000000
0.12000000
0.14000000
0.16000000
0.1
25
0.1
75
0.2
25
0.2
75
0.3
25
0.3
75
0.4
25
0.4
75
0.5
25
0.5
75
0.6
25
0.6
75
0.7
25
0.7
75
0.8
25
0.8
75
0.9
25
0.9
75
Pareto
vasicek
data
Figure 7a: Vasicek vs. Pareto vs. Data: R=0.25, PD-LGD Correlation=0.50
0.00000000
0.00500000
0.01000000
0.01500000
0.02000000
0.02500000
0.03000000
0.03500000
0.04000000
0.04500000
0.05000000
0.2
0.2
50.3
0.3
50.4
0.4
50.5
0.5
50.6
0.6
50.7
0.7
50.8
0.8
50.9
0.9
5
0.9
999
Pareto
vasicek
data
23
Once again, it is clear that in both figures the data imply a tail probability larger than that
predicted by the Vasicek distribution and closer to the higher tail probably predicted by
the Pareto distribution.
In short, in all variants of the full simulation model the tail of loss probability as
predicted by Pareto (a) exceeds the Vasicek predictions beyond a certain point and (b) fits
the data better. The threshold or cross-over point at which this “tail” result obtains is
found corresponds to PD values ranging from 1% to 2%, a range found in the mid to mid-
high risk segments for many banks’ portfolios.
5. Summary and Conclusion
In this paper we have estimated a Pareto distribution for loan losses as an
alternative to the commonly used Vasicek distribution. The advantage of using a Pareto
distribution is that it is free of the assumptions and restrictions underlying the Vasicek
distribution, but has similar Heavy-tailed properties. Using this distribution, we find a
higher probability of large losses than has been understood thus far. Our findings show
that the Pareto distribution predicts a higher likelihood of tail events and fits the data
more closely, when compared with the Vasicek distribution. The threshold or cross-over
point at which this “tail” result obtains seems to correspond to PD values ranging from
1% to 2%, a range found in the mid to mid-high risk segments for many banks’
portfolios.
Our findings points to a possible need to reconsider the modeling of
underlying systemic and idiosyncratic risk factors and examine the real possibility that
24
systemic risk and firm level risk might well interact, leading to a larger risk than a in
linear model in which systemic and firm idiosyncratic risk are assumed independent.
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