Filename

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

Filename

2

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%

Filename

3

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?

Filename

4

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)

Filename

5

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

Filename

6

, 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

Filename

7

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

Filename

8

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

Filename

9

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

Filename

10

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

Filename

11

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

Filename

12

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

Filename

13

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

Filename

14

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

Filename

15

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

Filename

16

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

Filename

17

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

Filename

18

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

Filename

19

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

Filename

20

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%

Filename

21

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

Filename

22

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%

Filename

23

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

Filename

24

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

Filename

25

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

Filename

26

Thank You!

http://www.econ.cam.ac.uk/faculty/pesaran/

Filename

27

Graveyard

Filename

28

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

Filename

29

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

Filename

30

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

Filename

31

-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