Firm Heterogeneity and Credit Risk
Diversification
Samuel G. Hanson
M. Hashem Pesaran
Harvard University
University of Cambridge and USC
Til Schuermann*
Federal Reserve Bank of New York,
Wharton Financial Institutions Center
Conference on Financial Econometrics
York, UK, June 2-3, 2006
* Any views expressed represent those of the authors only and not necessarily those of the Federal Reserve
Bank of New York or the Federal Reserve System.
Filename
Credit portfolio loss distributions
 We are primarily interested in generating (conditional)
credit portfolio loss distributions
N
N
i 1
i 1
lt 1   wi ,t li ,t 1 ,  wi ,t  1,
n
2
1
w

O
(
N
)
 i ,t
granularity condition
i 1
li ,t 1  I Vi ,t 1  Di ,t 1   LGDi ,t 1
= 100%
1
Filename
Obtaining credit loss distributions
 Credit loss distributions tend to be highly non-normal
– Skewed and fat-tailed
– Even if underlying stochastic process is Gaussian
– Non-normality due to nonlinearity introduced via the
default process
 Typical computational approach is through simulation for a
variety of modeling approaches
– Merton-style model
– Actuarial model
 Closed form solutions, desired by industry & regulators, are
often obtained assuming strict homogeneity (in addition to
distributional) assumptions
– Basel 2 Capital Accord
 What are the implications of imposing such homogeneity -or neglecting heterogeneity -- for credit risk analysis?
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Credit risk modeling literature
 Contingent claim (options) approach (Merton 1974)
– Model of firm and default process
– KMV (Vasicek 1987, 2002)
– CreditMetrics: Gupton, Finger and Bhatia (1997)
 Vasicek’s (1987) formulation forms the basis of the New Basel
Accord
– It is, however, highly restrictive as it imposes a number of
homogeneity assumptions
 A separate and growing literature on correlated default
intensities
– Schönbucher (1998), Duffie and Singleton (1999), Duffie
and Gârleanu (2001), Duffie, Saita and Wang (2006)
 Default contagion models
– Davis and Lo (2001), Giesecke and Weber (2004)
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Preview of results
 Our theoretical results suggest:
– Neglecting parameter heterogeneity can lead to
underestimation of expected losses (EL)
– Once EL is controlled for, such neglect can lead to
overestimation of unexpected losses (UL or VaR)
 Empirical study confirms theoretical findings
– Large, two-country (Japan, U.S.) portfolio
– Credit rating information (unconditional default risk: p)
very important
– Return specification important (conditional
independence)
 Under certain simplifying assumptions on the joint
parameter distribution, we can allow for heterogeneity with
minimal data requirements
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Filename
Firm returns and default: multi-factor
 Our basic multi-factor firm return process
ri , t 1  δi ft 1   i , t 1 ,
 i ,t 1 |  t ~ iidN (0,1); ft 1 |  t ~ N (0, I m )
t denotes the information available at time t
 Note that the multi-factor nature of the process matters
only when the factor loadings di are heterogeneous
across firms
 Firm default condition
zi ,t 1  I  ri ,t 1  ai ,t 1 | t   Et  zi ,t 1   p i ,t 1
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Introducing parameter heterogeneity: random
 Parameter heterogeneity can be introduced through the
standard random coefficient model
θi  θ  v i , v i ~ iid  0,  vv 
 aa
θi   ai , δ  ', v i   via , v  ',  vv  
 ω aδ
'
i
'
iδ
where vi is independent of ft+1 and t+1
ω aδ 
δδ 
 Parameter heterogeneity is a population property and
prevails even in the absence of estimation uncertainty
 Could be the case for middle market & small business
lending where it would be very hard to get estimates of i
– Use estimates from elsewhere for  and vv
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Introducing simple heterogeneity: random
 For simplicity, consider single factor model
 EL for Vasicek fully homogeneous case
 a 
ELt 1  Pr d f t 1   i ,t 1  a |  t    
2 
 1 d 
2
d
Note:

1 d 2
 Heterogeneity is introduced through ai
a<0
ai  a  vi , vi ~ iidN  0, aa 
Can be thought of as heterogeneity in default
thresholds and/or expected returns
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EL  under parameter heterogeneity
 Now we can compute portfolio expected loss
(recall a < 0 typically)

a
ELt 1   
 1 d 2  
aa


 a 
  
2 

 1 d 

 Can also be obtained from Jensen’s inequality since
( x)  0 for x  0.
  ai  
 E  ai  

 E  


2
2 
 1 d 
  1 d 
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 Neglecting this source of heterogeneity results in
underestimation of EL
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Systematic and random heterogeneity
 Impact on loss variance under random heterogeneity
is ambiguous
– EL not constant
 It helps to control for/fix EL
 Can only be done by introducing some systematic
heterogeneity, e.g. firm types
 E.g. 2 types, H, L, such that pL < pH < ½
 Calibrate exposures to types such that EL is same as
in homogeneous case (need NH, NL → )
p  wH p H  wLp L
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Systematic and random heterogeneity
 Holding EL fixed

aH
EL  p  wH  
 1 d 2  
aa



aL
  wL 

 1 d 2  
aa






 Loss variance under homogeneity
Vhom  F p ,p ,    p 2
F p i ,p j ,     2  1 (p i ),  1 (p j ),  
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Loss variance (UL)  under parameter
heterogeneity, for a given EL
 Loss variance under heterogeneity
Vhet  wH2 F p H ,p H ,    wL2 F p L ,p L ,  
 2 wH wL F p H ,p L ,    p 2

d2
1  d  aa
2


1  aa (1   )

 Theorem 1: Vhom > Vhet , assuming ELhom = ELhet
 Neglecting this source of heterogeneity results in
overestimation of loss variance
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Vhom > Vhet
 Proof draws on concavity of Fp, p, )
 Since    , F p ,p ,    F p ,p ,  
 Under p  wH p H  wLp L
F p ,p ,    wH2 F p H ,p H ,    wL2 F p L ,p L ,    2wH wL F p H ,p L ,  
 Concavity:
F p ,p ,    F  wH p H  wLp L ,p ,  
 wH F p H ,p ,    wL F p L ,p ,  
F p H ,p ,    wH F p H ,p H ,    wL F p H ,p L ,  
F p L ,p ,    wH F p L ,p H ,    wL F p L ,p L ,  
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Loss variance (UL)  under parameter
heterogeneity, for a given EL
 Holding EL fixed, neglecting parameter heterogeneity
results in the overestimation of risk
 Intuition: parameter heterogeneity across firms
increases the scope for diversification
 Relies on concavity of loss distribution in its
arguments
 Easily extended to many types, e.g. several credit
ratings
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Empirical application
 Two countries, U.S. and Japan, quarterly equity returns,
about 600 U.S. and 220 Japanese firms
 10-year rolling window estimates of return specifications
and average default probabilities by credit grade
– First window: 1988-1997
– Last window: 1993-2002
 Then simulate loss distribution for the 11th year
– Out-of-sample
– 6 one-year periods: 1998-2003
 To be in a sample window, a firm needs
– 40 consecutive quarters of data
– A credit rating from Moody’s or S&P at end of period
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Filename
Merton default model in practice
 Approach in the literature has been to work with market and
balance sheet data (e.g. KMV)
– Compute default threshold using value of liabilities from
balance sheet
– Using book leverage and equity volatility, impute asset
volatility
 We use credit ratings in addition to market (equity) returns
– Derive default threshold from credit ratings (and thus
incorporate private information available to rating
agencies)
– Changes in firm characteristics (e.g. leverage) are
reflected in credit ratings
 We use arguably the two best information sources available
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– Market
– Rating agency
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Modeling conditional independence
 The basic factor set-up of firm returns assumes that,
conditional on the systematic risk factors, firm returns
are independent
 A measure of conditional independence could be the
(average) pair-wise cross-sectional correlation of
residuals (in-sample)
 Similarly, we can measure degree of unconditional
dependence in the portfolio
– (average) pair-wise cross-sectional correlation of
returns (in-sample)
 Broadly, a model is preferred if it is “closer” to
conditional independence
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Model specifications
Models Descriptions
Return Specification
I
II
Vasicek
ri , t 1     rt 1  ui , t 1
Vasicek + Rating ri , t 1     rt 1  ui , t 1
III
CAPM
ri , t 1  i  i rc 1  ui , t 1
IV
V
CAPM + Sector
PCA
ri , t 1  i  1, i rt 1   2, i rj , t 1  ui , t 1
ri , t 1  i  i f t 1  ui , t 1
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Modeling conditional independence: results
Average Pair-wise
Correlation of Returns
Sample
Window
US&JP
US
1988-1997
0.1937
0.1933
# of firms
839
628
1993-2002
0.1545
0.1999
# of firms
818
600
JP
Model
Specifications
US&JP
US
JP
0.0222
0.0951
0.4217
III. CAPM
0.0218
0.0797
0.3868
IV. CAPM + Sector
0.0147
0.0711
0.3869
V. PCA
-0.0001
0.0016
0.0037
0.0549
0.1098
0.3332
0.0569
0.1157
0.3488
IV. CAPM + Sector
0.0439
0.1099
0.3543
V. PCA
-0.0008
-0.0006
0.0001
0.6011 I. Vasicek
211
Average Pair-wise Correlation of
Residuals
0.4191 I. Vasicek
218 III. CAPM
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Impact of heterogeneity: asymptotic portfolio
 Calibrate using simple 1-factor (CAPM) model
– Compare Vasicek (homogeneity), Vasicek + rating
(heterog. in default threshold/unconditional p)
Sample
1988-1997
1991-2000
1993-2002
Simulation
Year
Model
1998
2001
2003
VaR
EL
UL
99.0%
99.9%
I. Vasicek
1.23%
1.40%
6.82%
11.87%
II. Vasicek+Rating
1.23%
0.82%
4.11%
6.16%
III. CAPM
1.23%
0.52%
3.22%
5.30%
I. Vasicek
2.28%
1.65%
8.10%
12.07%
II. Vasicek+Rating
2.28%
0.91%
5.06%
6.58%
III. CAPM
2.28%
0.89%
5.31%
7.37%
I. Vasicek
3.26%
2.38%
11.61%
17.11%
II. Vasicek+Rating
3.26%
1.23%
6.94%
8.88%
III. CAPM
3.26%
0.95%
6.54%
8.84%
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Finite-sample/empirical loss distribution (2003)
40
III - CAPM
Models
IV - CAPM + Sector
35
I - Vasicek
II - Vasciek + Rating
V - PCA
30
III - CAPM
II - Vasicek + Rating
IV - CAPM + Sector
density
25
V - PCA
20
15
10
I - Vasicek
5
0
20%
18%
16%
Loss (% of Portfolio)
14%
12%
10%
8%
6%
4%
2%
0%
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Impact of heterogeneity: finite-sample portfolio
 Include multi-factor models
– Conditional independence?
Sample
1988-1997
Simulation
Year
Model
1998
EL  1.23%
1993-2002
2003
EL = 3.26%
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UL
99.9% VaR
I. Vasicek
1.47%
12.05%
II. Vasicek+Rating
1.07%
6.72%
III. CAPM
0.86%
5.56%
IV. Sector CAPM
0.88%
5.58%
V PCA
1.08%
7.69%
I. Vasicek
2.48%
17.47%
II. Vasicek+Rating
1.51%
9.46%
III. CAPM
1.27%
9.21%
IV. Sector CAPM
1.28%
9.20%
V PCA
1.51%
11.15%
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Calibrated asymptotic loss distribution (2003)
70
60
50
density
Vasicek
40
Vasicek+Rating
CAPM
30
20
10
0
0%
Filename
5%
10%
Loss (% of Portfolio)
15%
20%
22
Finite-sample/empirical loss distribution (2003)
40
III - CAPM
Models
IV - CAPM + Sector
35
I - Vasicek
II - Vasciek + Rating
V - PCA
30
III - CAPM
II - Vasicek + Rating
IV - CAPM + Sector
density
25
V - PCA
20
15
10
I - Vasicek
5
0
20%
18%
16%
Loss (% of Portfolio)
14%
12%
10%
8%
6%
4%
2%
0%
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Concluding remarks
 Firm typing/grouping along unconditional probability of default
(PD) seems very important
– Can be achieved using credit ratings (external or internal)
– Within types, further differentiation using return parameter
heterogeneity can matter
 Neglecting parameter heterogeneity can lead to
underestimation of expected losses (EL)
 Once EL is controlled for, such neglect can lead to
overestimation of unexpected losses (UL or VaR)
 Well-specified return regression allows one to comfortably
impose conditional independence assumption required by
credit models
– In-sample easily measured using correlation of residuals
– Measuring and evaluating out-of-sample conditional
dependence requires further investigation
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Thank You!
http://www.econ.cam.ac.uk/faculty/pesaran/
25
Filename
Graveyard
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Filename
Portfolio loss in Vasicek model
 Vasicek (1987) among first to propose portfolio
solution
 Loans are tied together via a single, unobserved
systematic risk factor (“economic index”) f and same
correlation 
ri   f  1    i ; f ,  i ~ iidN (0,1)
 Then, as N  , the loss distribution converges to a
distribution which depends on just p and 
– These two parameters drive the shape of the loss
distribution
– With equi-correlation and same probability of
default, default thresholds are also the same for
all firms
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Our contribution: conditional modeling and
heterogeneity
 The loss distributions discussed in the literature typically
do not explicitly allow for the effects of macroeconomic
variables on losses. They are unconditional models.
– Exception: Wilson (1997), Duffie, Saita and Wang
(2006)
 In Pesaran, Schuermann, Treutler and Weiner (JMCB,
forthcoming) we develop a credit risk model conditional
on observable, global macroeconomic risk factors
 In this paper we de-couple credit risk from business cycle
variables but allow for
– Different unconditional probability of default (by rating)
– Different systematic risk sensitivity across firms
(“beta”)
– Different error variances across firms
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Introducing heterogeneity
 Allowing for firm heterogeneity is important
– Firm values are subject to specific persistent effects
– Firm values respond differently to changes in risk
factors (“betas” differ across firms)
• Note this is different from uncertainty in the
parameter estimate
– Default thresholds need not be the same across firms
• Capital structure, industry effects, mgmt quality
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 But it [heterogeneity] gives rise to an identification problem
– Direct observations of firm-specific default probabilities
are not possible
– Classification of firms into types or homogeneous
groups would be needed
– In our work we argue in favor of grouping of firms by
29
their credit rating: pR
EL is under-estimated
p-H
p*
p
p-L
-3.5
-3
-2.5
-2
DD-L
Filename
-1.5
DD
-1
DD-H
-0.5
0
0.5
30