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Firm Heterogeneity and Credit Risk Diversification. 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. - PowerPoint PPT Presentation
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Firm Heterogeneity and Credit Risk Diversification
* 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.
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 EconometricsYork, UK, June 2-3, 2006
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Credit portfolio loss distributions
1 , , 1 ,1 1
2 1,
1
, 1 , 1 , 1 , 1
, 1,
( ) granularity condition
I
N N
t i t i t i ti i
n
i ti
i t i t i t i t
l w l w
w O N
l V D LGD
We are primarily interested in generating (conditional) credit portfolio loss distributions
= 100%
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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: )
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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, 1 1 , 1
, 1 1
,
| ~ (0,1); | ~ ( , )
i t i t i t
i t t t t m
r
iidN N
δ f
f 0 I
Our basic multi-factor firm return process
t denotes the information available at time t
Firm returns and default: multi-factor
Note that the multi-factor nature of the process matters only when the factor loadings i are heterogeneous across firms
Firm default condition
, 1 , 1 , 1 , 1 , 1|i t i t i t t t i t i tz I r a E z
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Introducing parameter heterogeneity: random
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
' '
, ~ ,
, ', , ',
i i i
aa ai i i i ia i vv
a
iid
a v
vv
δδ
δ δδ
θ θ v v 0ω
θ δ v vω
Parameter heterogeneity can be introduced through the standard random coefficient model
where vi is independent of ft+1 and t+1
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Introducing simple heterogeneity: random
Heterogeneity is introduced through ai
Can be thought of as heterogeneity in default thresholds and/or expected returns
, ~ 0,i i i aaa a v v iidN a < 0
For simplicity, consider single factor model
EL for Vasicek fully homogeneous case
Note:
1 1 , 1 2Pr |
1t t i t t
aEL f a
2
21
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EL under parameter heterogeneity
Now we can compute portfolio expected loss (recall a < 0 typically)
1 2 21 1t
aa
a aEL
2 21 1
i iE a a
E
Can also be obtained from Jensen’s inequality since for
( ) 0x 0.x
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 L < H < ½
Calibrate exposures to types such that EL is same as in homogeneous case (need NH, NL→ )
H H L Lw w
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Systematic and random heterogeneity
2 21 1H L
H L
aa aa
a aEL w w
Holding EL fixed
2hom
1 12
, ,
, , ( ), ( ),i j i j
V F
F
Loss variance under homogeneity
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Loss variance (UL) under parameter heterogeneity, for a given EL
Loss variance under heterogeneity
2 2het
2
, , , ,
2 , ,
H H H L L L
H L H L
V w F w F
w w F
2
21 1 (1 )aa aa
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)
, , , , ,F F Since
, , , ,
, , , ,
H H L L
H H L L
F F w w
w F w F
Concavity:
2 2, , , , , , 2 , ,H H H L L L H L H LF w F w F w w F
Under H H L Lw w
, , , , , ,
, , , , , ,
H H H H L H L
L H L H L L L
F w F w F
F w F w F
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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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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– 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
, 1 1 , 1
, 1 1 , 1
, 1 1 , 1
, 1 1, 1 2, , 1 , 1
, 1
I Vasicek
II Vasicek + Rating
III CAPM
IV CAPM + Sector
V PCA
i t t i t
i t t i t
i t i i c i t
i t i i t i j t i t
i t i
r r u
r r u
r r u
r r r u
r
Models Descriptions Return Specification
1 , 1i t i tu f
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Modeling conditional independence: results
Average Pair-wise Correlation of Returns
Average Pair-wise Correlation of Residuals
Sample Window US&JP US JP
Model Specifications US&JP US JP
1988-1997 0.1937 0.1933 0.6011 I. Vasicek 0.0222 0.0951 0.4217
# of firms 839 628 211 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
1993-2002 0.1545 0.1999 0.4191 I. Vasicek 0.0549 0.1098 0.3332
# of firms 818 600 218 III. CAPM 0.0569 0.1157 0.3488
IV. CAPM + Sector 0.0439 0.1099 0.3543
V. PCA -0.0008 -0.0006 0.0001
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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 )
VaR
Sample Simulation
Year Model EL UL 99.0% 99.9%
1988-1997 1998 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%
1991-2000 2001 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%
1993-2002 2003 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)
0
5
10
15
20
25
30
35
40
0% 2% 4% 6% 8% 10%
12%
14%
16%
18%
20%
Loss (% of Portfolio)
den
sity
I - Vasicek
II - Vasciek + Rating
III - CAPM
IV - CAPM + Sector
V - PCA
Models
I - Vasicek
II - Vasicek + Rating
V - PCA
III - CAPM IV - CAPM + Sector
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Impact of heterogeneity: finite-sample portfolio
Include multi-factor models– Conditional independence?
Sample Simulation
Year Model UL 99.9% VaR
1988-1997 1998 I. Vasicek 1.47% 12.05%
EL 1.23% 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%
1993-2002 2003 I. Vasicek 2.48% 17.47%
EL = 3.26% 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)
0
10
20
30
40
50
60
70
0% 5% 10% 15% 20%
Loss (% of Portfolio)
den
sity
Vasicek
Vasicek+Rating
CAPM
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Finite-sample/empirical loss distribution (2003)
0
5
10
15
20
25
30
35
40
0% 2% 4% 6% 8% 10%
12%
14%
16%
18%
20%
Loss (% of Portfolio)
den
sity
I - Vasicek
II - Vasciek + Rating
III - CAPM
IV - CAPM + Sector
V - PCA
Models
I - Vasicek
II - Vasicek + Rating
V - PCA
III - CAPM IV - CAPM + Sector
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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/
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Graveyard
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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
1 ; , ~ (0,1)i i ir f f iidN
Then, as N , the loss distribution converges to a distribution which depends on just 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
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
their credit rating: R
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-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5
EL is under-estimated
DD-L DD-H
-L
-H
*
DD