The Applicationof Gibbs Sampler Method to the Distribution of Asset Return Correlation in the New Basel Accord

The Application of Gibbs Sampler Method to the Distribution of Asset Return Correlation in the

New Basel Accord

鍾麗英

Lyinn Chung 國立台北大學統計學系副教授

余信萱 Hsin-Hsuan Yu 國立台北大學統計學系研究生

李 詩政 Shih-Cheng Lee 元智大學會計學副教授

Abstract

The asset correlation is the key variable for calculating the regula-

tory capital in the IRB approach of New Basel Accord. However, the

range of asset correlation has suffer a lot of controversy since the Sec-

ond Consultative paper (CP2) in 2001. Then, the third Consultative

paper (CP3) announced in April, 2003 formally introduced the asset

return correlation as a decreasing function of default probability. CP3

not only defined that the range of asset correlation from 0.12 to 0.24

for corporations but also addressed a negative relationship between as-

set correlation and probability of default. As CP3 did not explain the

theoretical concept for the formula of asset correlation, there were a

lot of studies discuss the appropriateness of this formula. This study

applied Bayesian method to ASRF model to estimate the posterior

distribution of asset correlation. We use the data for the firms in the

United State from year 2001 to 2005. The empirical results suggest

that the asset correlation is a decreasing function of probability of de-

fault and an increasing function of firm size, and indicate that there

may be some important factors impact the asset correlation were ig- nored.

Keywords: New Basel Accord, asset correlation, one-factor model, Bayesian

1 Introduction

In order to cope with the keen competition of international financial institution and maintain the soundness and stability of the interna- tional bank system, the supervisor institutions of various countries es- tablished a lot of capital controlled regulation to supervise risk, among them the “Basel Capital Accord” is the most representative.

In 1988, the Basel Committee for Banking Supervision announces the

“Basel Capital Accord”, that is Basel I, which defined credit risk and the minimum amount of capital that should be held by the bank. The accord required that banks must hold a minimum of 8% capital. Be- sides, in 1996, the Basel Committee incorporated market risk into the framework. This accord was ratified by the G-10 countries and is still used today to define the minimum amount of capital a bank must hold to cover loss arising from obligor default.

Since 1988, the framework contained in Basel I has been progressively

introduced not only in member countries but also in virtually all other

countries with active international banks. However, the shortcoming

of Basel I have been found gradually, and Committee pay much atten-

tion to the claim of improving the accord. In June 1999, the Commit-

tee issued a proposal , A New Capital Adequacy Framework (CP1),

to replace Basel I. Following extensive communication with banks and

industry groups, the final framework was issued in June 2004 and is

known as Basel II.

Asset correlation is an important component of the Basel II Accord for regulatory capital requirements of credit risk portfolios. It means the correlation of a given firm’s assets with the risk factor that summa- rizes general economic conditions and is a key parameter to determine the shape of the risk weight formulas that is based on the Asymptotic Single Risk Factor (ASRF) model. In the Basel Committee on Bank- ing Supervision (BCBS) document of January 2001 (BCBS, 2001a), asset correlations were assigned a value of 0.2 for all obligors. That is, the asset values of every obligor were assumed to have a factor load- ing of √

0.2 with the common risk factor. However, the fixed value cause much criticism and the range of asset correlation have revised two times, the fist time is 0.1 to 0.2 and now the range for the second time, ignoring the adjustment for the size of company, is between 0.12 and 0.24.

As Basel Committee did not explain the theoretical concept for the

formula of asset correlation, there were many researchers discussing

the appropriateness of this formula from different aspects. This study

applied Bayesian method to ASRF model in order to discuss the prob-

ability distribution of asset return correlation, and the relationship

among probability of default, firm size and asset correlation. Bayesian

inference is the process of fitting a probability model to a set of data

and summarizing the result by a probability distribution on the pa-

rameters of the model. By Bayesian method we can combine the

prior information with data to make plausibility statements or infer-

ences about parameters. However, alternative model can differ in the

specification of the prior distribution, in the specification of the like-

lihood, or both. Sensitivity analysis can be used to assess the effect of alternative prior distribution on the posterior inference. The basic method of sensitivity analysis is to fit several probability models to the same problem. We relaxed the restrictions of the distribution of the factor in one-factor model and investigated three kinds of prior distribution to make sure the sensitivity of the model. The analysis focused on firms in the United States with Standard Industrial Clas- sification (SIC) codes between 2000 and 5999, covering from year2001 to 2005.

Our empirical results confirm that the asset correlation is a decreasing function of probability of default, as suggested by BCBS(2001b,2002), and an increasing function of firm asset size, agreeing with what de- scribed in BCBS(2004). At the same time, our results indicate that there may be some important factors impact the asset correlation were not considered in the formula of asset correlation.

The paper is organized as follows. Chapter 2 summarizes the IRB

approach and also provides several empirical evidences on the asset

correlation. The framework of ASRF model, bayesian method and

Gibbs sampler on ASRF model will be described in chapter 3. Chap-

ter 4 presents the results of the posterior distribution of asset correla-

tion with by Z-score and firm size categories, respectively. Chapter 5

summarizes the results.

2 Literature review

2.1 Overview of Basel II

The objective of Basel II is to modernize the existing capital require- ments framework to make it more comprehensive and risk-sensitive, taking account of many modern financial institutions’ thorough risk management practices. A key aspect of the new framework is its flexibility. It provides institutions with the opportunity to adopt the approaches most appropriate to their situation and to the sophisti- cation of their risk management. In fact, the new capital adequacy scheme is based on three pillars:

1. minimum capital requirements;

2. supervisory review process;

3. market discipline.

With regard to the first pillar, the Committee proposes two approaches.

The first, called standardized approach, adopts external ratings, such

as those provided by rating agencies, export credit agencies, and other

qualified institutions. The second approach opens to internal rating

systems developed by banks, this is so called IRB approach, subject

to the meeting of specific criteria yet to be defined, and to validation

by the relevant national supervisory authority. The internal rating

approach also confers varying degrees of autonomy to banks in the

estimate of the parameters determining risk weightings, and conse-

quently, capital requirements: The foundation approach entails less

autonomy, and the advanced approach a greater one.

2.2 IRB approach

The approach based on internal ratings represents an important change in the Basel Committee’s principles for the determination of regula- tory capital requirements. The amount of capital required to support the economic risks of banking activities is calculated using the internal rating systems of banks. The IRB system requires four basic inputs, both in the foundation and advanced approach. In the IRB approach, institutions are allowed to determine the borrowers’ probabilities of default(PD), while those using the advanced IRB approach will also be permitted to use their own estimates of loss given default(LGD), exposure at default(EAD) and maturity(M). The four factors men- tioned above correspond to the risk parameters on which the Basel II IRB approach is built:

Probabilities of default (PD)

All banks whether using the foundation or the advanced methodology have to provide an internal estimate of the PD associated with the borrowers in each borrower grade. Probability of default per rating grade gives the average percentage of obligors that default in this rat- ing grade under normal business conditions.

Exposure at default (EAD)

Which gives an estimate of the amount outstanding in cases the bor- rower default. In most cases, EAD is equal to the nominal amount of the exposure but for certain exposures - e.g. those with undrawn commitments - it includes an estimate of future lending prior to de- fault.

Loss given default (LGD)

Loss given default rate, which is equal to one minus the recovery rate,

gives the percentage of exposure the bank might lose in case the bor-

rower default. Note that, average loss over long period of time under- state LGD rates during an economic downturn. Therefore, it needs to be adjusted upward to appropriately reflect adverse entomic condi- tions.

Maturity (M)

Where maturity is treated as an explicit risk component, like in the advanced approach, banks are expected to provide supervisors with the effective contractual maturity of their exposures.

2.3 Derivation of the Capital requirement

The capital requirements of banking books are derived from risk weight formulas, which were developed from Asymptotic Risk Factor (ASRF) model. It is believed that the precursor to the formula is the working paper of (Gordy, 2003). In the ASRF models, Z it is assumed the normalized asset return for firm i at time t. This variable can be decomposed in the following way:

Z _it = √

ρX _t + p

1 − ρ _it (1)

where

X _t ∼ N (0, 1)  _it ∼ N (0, 1)

The component  _it represents the risk specific to institution i, and X _t represents a common factor to all firms in the portfolio. √

ρ is the factor loading of the systematic risk and is often interpreted as the sensitivity to systematic. Under the assumption of distribution N ormal(0, 1), the correlation between the normalized asset return of any two borrower is ρ.

If a borrower’s return falls short of some threshold c, i.e.

Z _it < c ⇔ Y _it = 1 i = 1, ..., N _t , t = 1, ...T (2)

Where Y _it is an indicator variable and we assume that Y = 1 indicates that the firm defaults.

The unconditional probability of default as shown in (Vasicek, 1997) is as the form:

P D _i = P (Y _i = 1)

= P (Z _it < c)

= P ( √

ρX _t + p

1 − ρ _it < c)

= Φ(c)

Conditional on a realization x of the common random factor at time t then becomes the conditional probability of default:

P (Y _it = 1|X _t = x) = P ( _it < c − √

√ ρx

1 − ρ |X _t = x)

(3)

= Φ( c − √

√ ρx

1 − ρ |X _t = x)

P (Y _it = 1|X _t = x _99.9 ) = Φ( c − √

ρ Φ ⁻¹ (0.001)

√ 1 − ρ )

where X denoted the systematic risk factor, x _99.9 is the 99.9 ^th per- centile of the systematic risk factor, meaning that a worse outcome of the systematic risk factor only has a 0.01 percent change. and the conditional expected loss is:

E[L i |X _t = x 99.9 ] = LGD · Φ( c − √

ρ Φ ⁻¹ (0.001)

√ 1 − ρ )

(4)

= LGD · Φ( Φ ⁻¹ (P D _i ) − √

ρ Φ ⁻¹ (0.001)

√ 1 − ρ )

As mentioned in BCBS(2004a), the sum of expected loss(EL) and

unexpected loss(UL) are derived from the exposure’s conditional ex-

pected loss(CEL) by the formula

U L + EL = E[L _i |X _t = x _99.9 ]

(5)

= LGD · Φ( Φ ⁻¹ (P D _i ) − √

ρ Φ ⁻¹ (0.001)

√ 1 − ρ )

Finally, in Basel II the calibration of the risk weight only to unexpected losses. Thus the formula of capital requirement given by BCBS(2004b) is of the form:

K = LGD[Φ( Φ ⁻¹ (P D _i ) − √

ρ Φ ⁻¹ (0.001)

√ 1 − ρ ) − P D] × (6)

1 − b(P D)(M − 2.5) 1 − 1.5b(P D) Where

U L = LGD[Φ( Φ ⁻¹ (P D _i ) − √

ρ Φ ⁻¹ (0.999)

√ 1 − ρ )] − LGD · P D

= (U L + EL) − EL and

1 − b(P D)(M − 2.5) 1 − 1.5b(P D)

is full maturity adjustment as function of PD and M, with b(P D) = (0.11852 − 0.05478 × log(P D)) ²

means the smoothed (regression) maturity adjustment (smoothed over PDs).

The risk-weighted assets and capital requirements are related in a straightforward manner and the resulting formula is the following:

RW A _i = K(P D, ρ) · 12.5 · EAD (7)

The factor 12.5 compensate for the solvability coefficient of 0.08, and we can see that the asset correlation finally determine the shape of the risk weight formulas.

2.4 Asset Correlation in the IRB Approach

The asset correlation of Basel risk weight for corporate, sovereign and interbank exposures is given by

ρ i = 0.12 · 1 − e ^{−50P D}

1 − e ⁻⁵⁰ + 0.24 · (1 − 1 − e ^{−50P D}

1 − e ⁻⁵⁰ ) (8)

For corporate borrowers, the correlations ρ are first computed by Equation (8) and then modified as follows

ρ − 0.04, S _i ≤ EU R

ρ − 0.04 × (1 − (S _i − 5)/45), 5EU R < S _i ≤ 50EU R ρ, S i > 50EU R :

Where the S i is annual sales for firm i and 0.04 · (1 − ^S

₄₅ ⁻⁵ ) is size adjustment between 5 m EUR and 50 m EUR. Firms in the corpo- rate portfolio with a yearly turnover below 50 m EUR are treated as SMEs. They receive a capital relief dependent on their firm size which is measured by yearly turnover. As a consequence of the combined de- pendency on PD and firm size, the asset correlation for SMEs varies between 0.08 and 0.24.

The Basel Committee has also provided specific mappings between

probability of default p and asset correlation ρ for retail portfolios,

the formulas are as follows:

ˆ Residential mortgages

ρ _i = 0.15

ˆ Qualifying revolving retail exposures

ρ _i = 0.04

ˆ Other retail exposures

ρ i = 0.03 · 1 − e ^35p

1 − e ⁻³⁵ + 0.16 · (1 − 1 − e ^−35p

1 − e ⁻³⁵ )

The asset correlation function implies two systematic dependencies:

1. Asset correlation decrease with increasing PDs 2. Asset correlation increase with firm size.

The two empirical observations are the same as (Lopez, 2004)

2.5 Literature about Asset Correlation

Lopez (2004) In this paper, the ASRF approach was imposed on the

KMV methodology for determining credit risk capital charges in

order to examine the relationship between average asset corre-

lation, firm probability of default and firm asset size measured

by the book value of assets. Using data from year-end 2000,

credit portfolios consisting of US, Japanese, and European firms

are analyzed. Subportfolios based on either firm default proba-

bility or asset size were constructed for the univariate analysis,

and subportfolios based on both variables were used for the bi-

variate analysis. The empirical results suggest that average asset

correlation is a decreasing function of probability of default and

increasing function of firm size. The result suggest that these

factors may need to be accounted for in the regulatory capital requirements for credit risk.

Dietsch and J.Petey (2004) use a one-factor credit risk model to provide new estimates of stationary default probabilities and asset correlations for SMEs in the German and French market and conclude that the relationship between PDs and asset correlations is not negative but positive, especially at the industry level, in the two countries. It is also possible to distinguish different segments inside the SMEs’ population: at least between very small and small SMEs and large SMEs. However, their evidence support for a decrease of asset correlation with firm size. Their results show that the asset correlations in SME population are very week (1-3%) on average and decrease with size.

D¨ ullmann and Scheule (2003) The asset correlation is estimated from time series of ten years with default histories of 53280 Ger- man companies. The sample is divided into categories that are homogenous with respect to default probability (PD) and firm size. In order to estimate the model parameters in the ASRF model we apply three different estimation methods: maximum likelihood estimator (MLE), full-information maximum likelihood estimator (FIML) and method of moments estimator (MME) three kinds of methods to estimate the relationship between asset correlation, probability of default and firm size. Finally, they find that the asset correlation overall increases with firm size but do not observe clear relation between asset correlation and probabil- ity of default.

R¨ osch (2002) examines the effect of the business cycle on correla-

tion in a one factor model. He use database of German corporate

bankruptcies and find that correlations are throughout lower than assumed in the New Basel Accord. Though he does not specifi- cally address the issue of the relationship between probability of default and asset correlation, his findings are still relevant at this point. This is because in some sectors he includes lagged values of the default rates for the respective sector. As described before, the estimates for the parameters of these variables can be inter- preted as explaining parts of the correlation with the common risk factor. When compared to the other included regression variables for a certain sector, the lagged default rates showed smaller stan- dard errors and were all positive. This strongly contradicts the Basel-II assumption of a decreasing asset correlation with PD.

Hamerle and R¨ osch (2003) have made strong use of additional

explanatory variables within a model. It turns out that the in-

clusion of these regression variables reduce the estimates for asset

correlation significantly. To give an example: They use default

data from the U.S market and the rating system of Standard and

Poor’s. For the BB grade the model without regression variables

produces an estimate of 6.04% compared to a 0.67% when the

Federal Reserve Funds Rate is included as a one year lagged vari-

able. This enormous increase in precision can be explained by the

fact, that the business cycle is reflected in macroeconomic quan-

tities. Therefore, the inclusion of these figures will increase the fit

of the model drastically. The given estimates for asset correlation

are non-monotone in the rating categories and do therefore not

support the Basel-II assumptions.

3 Research Method

3.1 Bayesian Statistical Analysis

Bayesian inference is the process of fitting a probability model to a set of data and summarizing the result by a probability distribution on the parameters of the model. It is a coherent framework for com- bining “prior information” with data to make plausibility statements or inferences about parameters. The relationship between prior infor- mation and data set can be seen from the Bayes Rule:

p(θ|y) = p(θ, y)

p(y) = p(y|θ)p(θ)

p(y) = p(y|θ)p(θ) R

Θ p(θ)p(y|θ)dθ (9) Bayes’s Rule is often written as:

p(θ|y) ∝ p(θ)p(y|θ) i.e.

P osterior ∝ P rior × Likelihood

The posterior distribution can be seen as a compromise between the prior and the data. In general, this can be seen on the two well known relationships:

E[θ] = E[E[θ|y]] (10)

V ar(θ) = E[V ar(θ|y)] + V ar(E[θ|y]) (11) The Equation (10) means that the prior mean is the average of all possible posterior means, and the Equation (11) tells us that the pos- terior variance is, on average, smaller than the prior variance.

Take for a concrete example, a case where the prior is θ ∼ Beta(4, 3)

and the observation y=2 is from Bin(5, θ). The relationship between

prior, likelihood and posterior can be seen as Figure 1.It seems like that the posterior distribution may have a mean somewhere between the locations of the peaks of f p(y|θ) and p(θ), and this graphic indi- cates how prior and likelihood combine to produce the posterior.

Figure 1: Prior vs. Likelihood vs. Posterior

3.2 Gibbs Sampler

From Equation (9), we can see that to derive the posterior distribution

needs through the technique of integration. However, if the parame-

ter which we are interested in is a high dimension one, it must be a

stiff matter, that is why we use Gibbs sampler. The Gibbs sampler

is a technique for generating random variables from a distribution in-

directly, without having to calculate the density. Through the use

of techniques like the Gibbs sampler, we are able to avoid difficult

calculations. For Θ = (θ ₁ , θ ₂ , . . . , θ _k ), the fundamental idea of Gibbs

sampler is, from some starting vector Θ ⁽⁰⁾ one creates Θ ⁽ⁱ⁺¹⁾ from Θ ⁽ⁱ⁾

by “updating” in succession the k coordinates of Θ ⁽ⁱ⁾ drawing at ran-

dom from the conditional distribution of each variable given current

values of all the others.

In order to illustrate the basic idea, consider the simple example of Gibbs sampling from the joint discrete distribution of Θ = (θ ₁ , θ ₂ ) given in Table 1 below:

Table 1: pmf of Θ = (θ ₁ , θ ₂ )

θ 2 = 1 θ 2 = 2 θ ₁ = 1 0.2 0.1 θ ₁ = 2 0.2 0.2 θ ₁ = 3 0.1 0.2

This joint distribution has conditional distributions:

θ ₁ g(θ ₁ |θ ₂ = 1)

1 2/5

2 2/5

3 1/5

θ ₁ g(θ ₁ |θ ₂ = 2)

1 1/5

2 2/5

3 2/5

and

θ 2 g(θ 2 |θ 1 = 1)

1 2/3

2 1/3

θ 2 g(θ 2 |θ 1 = 2)

1 1/2

2 1/2

θ 2 g(θ 2 |θ 1 = 3)

1 1/3

2 2/3

Suppose that one beings Gibbs sampling with initial Θ ⁰ = (2, 1)

Then generates a replacement for θ ₁ = 2 by generating a value from g(θ ₁ |θ ₂ = 1), and suppose that θ ₁ = 3 is generates. Then one must generate a replacement for θ 2 = 1 using the distribution specified by g(θ ₂ |θ ₁ = 3). If θ ₂ is generated, after one complete Gibbs cycle we obtained:

Θ ¹ = (3, 2)

After generating a long string of Θ in this way, the theory of Markov Chains implies that the relative frequency distribution of the gener- ations will approximate the distribution in Table 1. (Besag, 1974) establish that this set is sufficient to determine the joint or any mar- gins distribution.

3.3 The Asymptotic Single Risk Factor Model

In the ASRF models, Z _it is assumed the normalized asset return for firm i at time t. This variable can be decomposed in the following way:

Z it = √

ρX t + p

1 − ρ it (12)

where

X _t ∼ N (µ _x = 0, σ _x ² = 1)  _it iid ∼ N (µ _ = 0, σ ² _ = 1)

and X _t is independent of  _it , which means that Z _it is considered to have a standardized Gaussian distribution.(i.e. µ _z = 0, σ _z ² = 1) The component  _it represents the risk specific to institution i, and X _t rep- resents a common factor to all firms in the portfolio. √

ρ is the factor loading of the systematic risk and often interpreted as the sensitivity to systematic, and ρ is the correlation between the normalized asset return of any two borrower.

Under ASRF model, the correlation between two borrowers i and j comes:

Corr(Z _i , Z _j ) = Cov(Z _i , Z _j ) pV ar(Z _i )pV ar(Z _j )

(13)

= E(Z _i Z _j ) − E(Z _i )E(Z _j )

σ _zi σ _zj

where

E(Z

Z

) = E[( √

ρX + p

1 − ρ

)( √

ρX + p

1 − ρ

)]

= E[ρX

+ √ ρ p

1 − ρX

+

√ rho p

1 − ρX

+ (1 − ρ)



]

= ρE(x

) + √ ρ p

1 − ρE(X)E(

) + √ ρ p

1 − ρE(X)E(

) + (1 − ρ)E(



)

= ρσ

+ ρµ

+ 2 √ ρ p

1 − ρµ

µ

+ (1 − ρ)µ

(14)

and

E(Z _i )E(Z _j ) = [ √

ρµ _x + p

1 − ρµ _ ] ²

= ρµ ² _x + 2 √ ρ p

1 − ρµ _x µ _ + (1 − ρ)µ ² _ (15)

From Equation (14) and (15) we can obtain the covariance of Z _i and Z _j

Cov(Z _i , Z _j ) = E(Z _i Z _j ) − E(Z _i )E(Z _j )

= ρσ _x ² + ρµ ² _x + 2 √ ρ p

1 − ρµ _x µ _ + (1 − ρ)µ ² _x − (ρµ ² _x + 2 √

ρ p

1 − ρµ _x µ _ + (1 − ρ)µ ² _ )

= ρσ _x ²

So the correlation between Z _i and Z _j becomes:

Corr(Z _i , Z _j ) = ρσ _x ²

σ _zi σ _zj (16)

Under the assumption of ASRF model σ _x ²

σ zi

σ zj all equal to one, so the correlation between any two borrowers is ρ, but if the distribution of systematic risk factor, X, and idiosyncratic risk factor, , are not standard normal distribution the asset correlation will not equal to ρ.

3.4 ASRF Model with Bayesian Method

In this paper, we relax the restrictions on the distribution of system-

atic risk factor and idiosyncratic risk factor and let them follow normal

distribution with parameters (µ _x , σ _x ² ) and (µ _ , σ _ ² ). Not alike the prop-

erties in frequency method that the parameter is a fixed variable, in

bayesian method all parameters could have their own distribution, and we assume µ _x and µ _ follow normal distribution with parameters (0,1), at the same time, σ _x ² and σ ² _ follow inverse Gamma(1, 1).

Because of the alteration in the distribution, the asset correlation be- tween two borrowers in Equation (16) is not ρ any more; however, we should adjust the value to solve the true asset correlation, and our ASRF model using Bayesian method becomes:

Z _it | µ _Z σ _Z ² ∼ N ormal(µ _Z , σ _Z ² ) X _t | µ _x σ _x ² ∼ N ormal(µ _x , σ _x ² )

 _it | µ _ σ _ ² ∼ N ormal(µ _ , σ _ ² ) where

µ _Z = √

ρµ _X + p

1 − ρµ _ σ _Z ² = ρσ _x ² + (1 − ρ)σ ² _

µ x ∼ N ormal(0, 1) µ _ ∼ N ormal(0, 1)

σ ² _x ∼ inv − Gamma(1, 1) σ _ ² ∼ inv − Gamma(1, 1)

ρ ∼ U nif orm(0, 1) The actual correlation becomes ρ _{T RU E} , not ρ

ρ _{T RU E} = Cov(Z _j , Z _k ) pV ar(Z j )pV ar(Z _k )

= ρ σ ² _x σ _Z ²

For the input data, we use the firm’s equity daily return as a proxy

of asset return, as mentioned in (Henneke, 2005), it has been industry

practice to use equity correlations as a proxy for asset correlations.

For systematic risk factor, we applied stock index returns to estimate, and for idiosyncratic risk factor, , the theory of CAPM is adopted.

We use no-intercept regression analysis and remain residuals as the idiosyncratic risk factor.

The cause of choosing prior distribution for µ _x and µ _ as N ormal is due to the properties of data, and inverse Gamma distribution for variance of systematic risk factor and idiosyncratic risk factor is be- cause of the positive nature of variance. Additionally, variance of all distribution follows Chi square distribution, which is a special case of Gamma distribution, and we take inverse Gamma but not Gamma is just a usual practice of Bayesian method. For the square of loading of systematic risk factor, ρ, we take three kinds of prior: U nif orm(0, 1), N ormal(0, 1)I(0, 1) ¹ and Beta(0.5, 0.5) to see if they have similar ten- dency. If they do have similar tendency, we can say that our likelihood belief is strong enough, so no matter under what kinds of prior, we can obtain same conclusion.

3.5 Posterior distribution of asset correlation

— Application of Gibbs Sampler

In this paper, we use Gibbs sampler to simulate a sequence to ap- proximate the posterior distribution. Through the use of techniques like Gibbs sampler, we are able to avoid difficult calculations. In the following content, we illustrate how to use Gibbs sampler to obtain the posterior distribution of asset correlation in our Bayesian ASRF model:

1

“I(0,1)” means we only focus on the values between 0 and 1

ˆ Generate initial values:

Θ ⁽⁰⁾ = {µ ⁽⁰⁾ _x , µ ⁽⁰⁾ _ , σ _x ^{2 (0)} , σ _ ^{2 (0)} , ρ ⁽⁰⁾ }

ˆ At time 1 derive Θ ⁽¹⁾ = {µ ⁽¹⁾ x , µ ⁽¹⁾  , σ x ^{2 (1)} , σ  ^{2 (1)} , ρ ⁽¹⁾ } from 1. draw µ ⁽¹⁾ x form p(µ _x |µ ⁽⁰⁾  , σ x ^{2 (0)} , σ  ^{2 (0)} , ρ ⁽⁰⁾ , Z, X, ) 2. draw µ ⁽¹⁾  form p(µ _ |µ ⁽¹⁾ x , σ x ^{2 (0)} , σ  ^{2 (0)} , ρ ⁽⁰⁾ , Z, X, ) 3. draw σ x ^{2 (1)} form p(σ _x ² |µ ⁽¹⁾ x , µ ⁽¹⁾  , σ  ^{2 (0)} , ρ ⁽⁰⁾ , Z, X, ) 4. draw σ  ^{2 (1)} form p(σ _ ² |µ ⁽¹⁾ x , µ ⁽¹⁾  , σ x ^{2 (1)} , ρ ⁽⁰⁾ , Z, X, ) 5. draw ρ ⁽¹⁾ form p(ρ|µ ⁽¹⁾ x , µ ⁽¹⁾  , σ x ^{2 (1)} , σ  ^{2 (1)} , Z, X, )

ˆ Repeat the simulation method until time t we will get a long string of Θ:

Θ ⁽¹⁾ = {µ ⁽¹⁾ _x , µ ⁽¹⁾ _ , σ _x ^{2 (1)} , σ _ ^{2 (1)} , ρ ⁽¹⁾ } Θ ⁽²⁾ = {µ ⁽²⁾ _x , µ ⁽²⁾ _ , σ _x ^{2 (2)} , σ _ ^{2 (2)} , ρ ⁽²⁾ }

...

Θ ^(t) = {µ ^(t) _x , µ ^(t) _ , σ ^{2 (t)} _x , σ _ ^{2 (t)} , ρ ^(t) } (17) It turns out that under reasonably general conditions, as t → ∞, the final observation in Equation (17), namely ρ ^(t) is effectively a sample point from f (ρ). So we discard any of the pre-convergence values from the summary, it is so called “burn-in”. To check convergence, we can exam the trace plots of the sample values versus iteration to look for evidence of when the simulation appear to have stabilized. The fol- lowing plots are example of: (i) chains for which convergence looks reasonable Figure 2(a); and (ii) chains which have clearly not reached convergence Figure 2 (b).

Once convergence has been achieved, we will need to run the simu-

lation for a further number of iterations to obtain samples that can

(a) (b)

Figure 2: convergence and not reached convergence

be used for posterior inference. The more samples, the more accurate will be the posterior estimates.

One way to assess the accuracy of the posterior estimates is by calcu- lating the Monte Carlo error for each parameter. This is an estimate of the difference between the mean of the sampled values (which we are using as our estimate of the posterior mean for each parameter) and the true posterior mean. As a rule of thumb, the simulation should be run until the Monte Carlo error for each parameter of interest is less than about 5% of the sample standard deviation.

3.6 Data Description

Our empirical investigation considers the cross-sectional data of in-

dustrial firms. We collect all data from the COMPUSTAT database

and the Center for Research in Securities Prices (CRSP) for firms on

the three main exchanges: the New York Stock Exchange (NYSE), the

American Stock Exchange (AMEX), and the NASDAQ Stock Market

in the United States covering the 5-year period from year 2001 to year

2005. Data include industrial firms with Standard Industrial Classifi-

cation (SIC) codes between 2000 and 5999 ( from “Food and Kindred

Products” to “Misc. Retail Stores, Nec” Firms in special industries

like mining, construction, financial and service, and firms with missing

or unreasonable values were excluded from analyses.

4 Empirical Results

As can be seen from Equation (2.8), Basel II assumes a downward sloping relationship between the asset correlation and the probability of default. Namely, the correlation between the assets of low-risk firms is assumed to be higher than the ones between high-risk firms. Several studies have results which did not support this relationship (Dietsch and Petey, 2004; D¨ ullmann and Scheule, 2003; Hamerle et al., 2003;

R¨ osch, 2002). This thesis explored the relationship between asset cor- relation and default probability using Bayesian method. However, the estimation of default probability is not available. Altman (1960,1990), a financial economist, developed a model for predicting the likelihood that a corporation would go bankrupt. This model uses five financial ratios that combine in a specific way to produce a single number. This number, called the Z-score, is a general measure of corporate financial health. Z-score is used as the proxy variable for default probability in this study.

The positive relationship between asset correlation and firm size is generally supported (Lopez, 2004; Dietsch and Petey, 2004; D¨ ullmann and Scheule, 2003). This thesis also explored this relationship using Bayesian method. Three different shapes of prior distributions were used to explore the posterior distribution of asset return correlations due to the lack of enough historical data or the experts’ suggestions.

These three priors are ”standard normal”, ”Beta(0.5,0.5)”, and ”Uni- form(0,1)”. The study results from these three priors are very similar.

The results for uniform prior are tabulated in the text and the results

of the other two priors are listed in appendices.

4.1 Based on Z-score categories

In this section we choose Altman’s Z-score as a proxy of probabil- ity of default, and classify all firms into three categories. The Z- score formula for predicting bankruptcy is a multivariate formula for a measurement of the financial health of a company and a powerful diagnostic tool that forecasts the probability of a company entering bankruptcy within a 2 year period. Studies measuring the effectiveness of the Z-score have shown the model is often accurate in predicting bankruptcy (72%-80% reliability).

The Z-score bankruptcy predictor combines five common business ra- tios, using a weighting system calculated by (Altman, 1968) to deter- mine the likelihood of a company going bankrupt.

The five ratios are:

. Working capital-to-total assets . Retained earnings-to-total assets

. Earnings before interest and taxes (EBIT)-to-total assets . Market value equity-to-book value of total debt

. Sales-to-total assets

The final discriminant function is as follows:

Z = 1.2X1 + 1.4X2 + 3.3X3 + 0.6X4 + 1.0X5 . X1 = working capital/total assets

. X2 = retained earnings/total assets

. X3 = earnings before interest and taxes/total assets . X4 = market value equity/book value of total liabilities . X5 = sales/total assets, and

. Z = overall Index or Score

If a financial risk manager uses a corporate borrower’s accounting ratio weighted by the estimated coefficients in the Z function resulting in a Z-score below a critical value ( for instance, in Altman’s initial study, 1.81 ), this debtor would be sorted as “bad” and the loan may be refused. The detailed boundary could be summarized as follows:

ˆ Z-score >2.99 means a healthy company

ˆ 1.81 < Z-score <2.99 represents it is in the gray zone

ˆ Z-score<1.81 implies that the firm is unhealthy

Within the ASRF model framework, we expect to see a negative re- lationship between firms’ asset correlations and their probabilities of default (PD). That is, as a firm’s PD increases due to its worsening condition and approaching possible default, it is reasonable to assume that idiosyncratic factors begin to take on a more important role rel- ative to the common, systematic risk factor.

Table 2 to Table 5present the summary statistics: posterior mean,

standard deviation, MC error, median and 95% credible interval, for

the distribution of asset correlation by Z-score category for year 2001

to year 2005, respectively.

Table 2: Asset correlation in year 2005 for Uniform prior by z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.0764 0.0035 0.0001 0.0696 0.0764 0.0832 1.81<Z-score<2.99 0.0554 0.0062 0.0001 0.0432 0.0555 0.0673 Z-score<1.81 0.0345 0.0062 0.0001 0.0225 0.0345 0.0466 The sample size of category “Z-score>2.99” is 1422, of category “1.81<Z-score<2.99”

is 454 and of category “Z-score<1.81” is 588.

Table 3: Asset correlation in year 2004 for Uniform prior by z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.0897 0.0028 0.0001 0.08416 0.08966 0.0953 1.81<Z-score<2.99 0.0828 0.0056 0.0001 0.0721 0.0829 0.0937 Z-score<1.81 0.0654 0.0064 0.0002 0.0530 0.0654 0.0781 The sample size of category “Z-score>2.99” is 1499, of category “1.81<Z-score<2.99”

is 520 and of category “Z-score<1.81” is 531.

Table 4: Asset correlation in year 2003 for Uniform prior by z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.0852 0.0040 0.0001 0.0774 0.0852 0.0931 1.81<Z-score<2.99 0.0756 0.00624 0.0001 0.0636 0.0756 0.0881 Z-score<1.81 0.0542 0.0096 0.0003 0.0359 0.0540 0.0736 The sample size of category “Z-score>2.99” is 1339, of category “1.81<Z-score<2.99”

is 490 and of category “Z-score<1.81” is 597.

Table 5: Asset correlation in year 2002 for Uniform prior by z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.1479 0.0048 0.0002 0.1386 0.1478 0.1576 1.81<Z-score<2.99 0.1239 0.0066 0.0002 0.1110 0.1239 0.137 Z-score<1.81 0.1066 0.0114 0.0005 0.0866 0.1058 0.1306 The sample size of category “Z-score>2.99” is 1112, of category “1.81<Z-score<2.99”

is 516 and of category “Z-score<1.81” is 783.

Table 6: Asset correlation in year 2001 for Uniform prior by z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.1665 0.0058 0.0002 0.1554 0.1664 0.1786 1.81<Z-score<2.99 0.0799 0.0073 0.0001 0.0656 0.0798 0.0945 Z-score<1.81 0.0944 0.0123 0.0005 0.0726 0.0935 0.1214 The sample size of category “Z-score>2.99” is 1209, of category “1.81<Z-score<2.99”

is 480 and of category “Z-score<1.81” is 607.

From Table 2 to Table 5 we can see that the positive relationship be- tween Z-score and asset correlation exist in year 2002 to year 2005;

however in year 2001, the results do not support the positive relation- ship. We found that asset correlation for category of < 1.81 is larger than that of category 1.81 to 2.99 but the asset correlation of category

> 2.99 is still the largest one among these three categories. All of the

“MC error”s are smaller than 5% of the standard deviation, and it is the thumb rule to assess the accuracy of the posterior estimates.

All of the posterior means from the above tables are displayed in Table

8. We can see that in the same Z-score category the asset correlation

is different from year to year.

Table 7: Sample size by Z-score category

year Z-score>2.99 1.81<Z-score<2.99 Z-score<1.81

2005 1422 454 588

2004 1499 520 531

2003 1339 490 597

2002 1112 516 783

2001 1209 480 607

Table 8: Means of asset correlation for year 2001-2005 by Z-score category for Uni- form prior

year Z-score>2.99 1.81<Z-score<2.99 Z-score<1.81

2005 0.0764 0.0554 0.0345

2004 0.0897 0.0828 0.0654

2003 0.0852 0.0756 0.0542

2002 0.1479 0.1239 0.1066

2001 0.1665 0.0799 0.0944

Figure 3 we present the tendency of asset correlation year by year,

and we can see a decline tendency roughly. Actually, the downward

tendency is most significant in Z-score>2.99 category. Furthermore,

from year 2002 to 2005, the range of shift is similar in all of the three

groups.

Figure 3: Means of asset correlation during years 2001-2005 by Z-score category

In the end, we will go a step further to exam whether under different

prior distribution, the posterior distribution of asset correlation will

be similar, if so, we can say that our data set is strong enough and will

not affect significantly due to the alteration of prior. All the Monte

Carlo Errors (MC errors) are less than 5%. The accuracy of the poste-

rior distribution of asset correlation is confirmed and not presented in

the following tables. Table 9 is the posterior mean of asset correlation

under N ormal(0, 1)I(0, 1) prior, and Table 10 is under Beta(0.5, 0.5)

prior. The detailed study results for prior Normal and Beta are given

in Appendix A. The study results indicate that no matter under what

kinds of prior, the posterior distribution of asset correlation is similar

to the one for uniform prior. Then in the next section, we will inves-

tigate the effect of different firm size on asset correlation.

Table 9: Means of asset correlation for year 2001-2005 by Z-score category for Nor- mal prior

year Z-score>2.99 1.81<Z-score<2.99 Z-score<1.81

2005 0.0764 0.0552 0.0342

2004 0.0937 0.0825 0.0652

2003 0.0851 0.0755 0.0546

2002 0.1475 0.1239 0.1068

2001 0.1657 0.0799 0.0935

Table 10: Means of asset correlation for year 2001-2005 by Z-score category for Beta prior

year Z-score>2.99 1.81<Z-score<2.99 Z-score<1.81

2005 0.0764 0.0546 0.0341

2004 0.0939 0.0824 0.0648

2003 0.0847 0.0754 0.0532

2002 0.1471 0.1238 0.1057

2001 0.1663 0.0795 0.0927

We list the results of Lopez(2004) in Figure 4 to make a comparision.

Lopez(2004) used data from year-end 2000 and invested the relation-

Figure 4: Results of Lopez(2004) for EDF category

ship of EDF, asset size and asset correlation. His results presented

that the asset correlation in the U.S. is between 0.15 and 0.225 based

on EDF category. Our results from year 2001 to 2005 are smaller than

Lopez’s results, but the negative relation between defalt probability

and asset correlation are similar.

4.2 Based on firm size categories

The second part of our analysis focuses on the relationship between firm size and asset correlation, and we use the book value of firm’s equity to estimate the firm’s size. As same as previous section, we group the firm size into three categories, the cut-off points follow the one used in Lopez’s paper (2004).

In the general theory of portfolio diversification, as the number of different securities within a portfolio increases, the portfolio becomes more diversified, and the idiosyncratic element of the portfolio’s return becomes less important. An analogous view could be taken with re- spect to a firm’s asset size; that is, as a firm becomes larger and comes to contain more assets, its risk and return characteristics should more closely resemble the overall asset market and be less dependent on the idiosyncratic elements of the individual business lines. Within the ASRF model framework, this intuition suggests that a firm’s asset correlation should increases as its asset size increases. From Table 11 to Table 17, the study results confirm the aforementioned intuition.

The MC errors are less than 5% and confirm the accuracy of posterior distribution of asset correlation.

Table 11: Asset correlation in year 2005 for Uniform prior by firm size category firm size mean sd MC error 2.5% median 97.5%

> $1b 0.1385 0.0047 0.0001 0.1292 0.1386 0.1477 ( $100m,$1b) 0.1310 0.0038 0.0001 0.1235 0.1310 0.1382 ( $0m,$100m) 0.0187 0.0054 0.0001 0.0082 0.0188 0.0290

The sample size of category “> $1b” is 567, of category “( $100m,$1b)” is 1025 and of

category “( $0m,$100m)” is 867.

Table 12: Asset correlation in year 2004 for Uniform prior by firm size category firm size mean sd MC error 2.5% median 97.5%

> $1b 0.1758 0.0045 0.0001 0.1669 0.1759 0.1843 ( $100m,$1b) 0.1642 0.0038 0.0001 0.1567 0.1642 0.1717 ( $0m,$100m) 0.0388 0.0056 0.0001 0.0277 0.0390 0.0498

The sample size of category “> $1b” is 601, of category “( $100m,$1b)” is 1041 and of category “( $0m,$100m)” is 901.

Table 13: Asset correlation in year 2003 for Uniform prior by firm size category firm size mean sd MC error 2.5% median 97.5%

> $1b 0.2136 0.0046 0.0001 0.2046 0.2137 0.2225 ( $100m,$1b) 0.1459 0.0041 0.0002 0.1378 0.1459 0.1544 ( $0m,$100m) 0.0295 0.0062 0.0001 0.0181 0.0294 0.0419

The sample size of category “> $1b” is 556, of category “( $100m,$1b)” is 989 and of category “( $0m,$100m)” is 874.

Table 14: Asset correlation in year 2002 for Uniform prior by firm size category firm size mean sd MC error 2.5% median 97.5%

> $1b 0.2771 0.0055 0.0002 0.2666 0.2772 0.2879 ( $100m,$1b) 0.2115 0.0049 0.0002 0.2020 0.2115 0.2213 ( $0m,$100m) 0.0482 0.0078 0.0002 0.0334 0.0480 0.0639

The sample size of category “> $1b” is 481, of category “( $100m,$1b)” is 958 and of

category “( $0m,$100m)” is 958.

Table 15: Asset correlation in year 2001 for Uniform prior by firm size category firm size mean sd MC error 2.5% median 97.5%

> $1b 0.2767 0.0057 0.0002 0.2656 0.2767 0.2881 ( $100m,$1b) 0.1920 0.0061 0.0003 0.1808 0.1917 0.2053 ( $0m,$100m) 0.0645 0.0089 0.0003 0.0484 0.0640 0.0829

The sample size of category “> $1b” is 455, of category “( $100m,$1b)” is 934 and of category “( $0m,$100m)” is 895.

Unlike the results of classification by Z-score in year 2001 that the cat- egory of Z-score under 1.81 is bigger than category of Z-score between 18.1 and 2.99, the results by firm size categories conclude that firm’s asset correlation increases as its asset size increases for year 2001 to year 2005. Besides, comparing the results under different prior from Table 17, Table 18 and Table 19, the results are very similar for three different priors. Therefore, our models by firm size category pass the sensitivity analysis and form a strong likelihood belief.

Table 16: Sample size–classified according to firm size year > $1b ( $100m,$1b) ($0m,$100m)

2005 567 1025 867

2004 601 1041 901

2003 556 989 874

2002 481 958 958

2001 455 934 895

Table 17: Means of asset correlation for year 2001-2005 by firm size category for Uniform prior

year > $1b ( $100m,$1b) ($0m,$100m)

2005 0.1385 0.1310 0.0187

2004 0.1758 0.1642 0.0388

2003 0.2136 0.1459 0.0295

2002 0.2771 0.2115 0.0482

2001 0.2767 0.1920 0.0645

Table 18: Means of asset correlation for year 2001-2005 by firm size category for Normal prior

year > $1b ( $100m,$1b) ($0m,$100m)

2005 0.1385 0.1309 0.0187

2004 0.1757 0.1640 0.0387

2003 0.2138 0.1461 0.0297

2002 0.2775 0.2111 0.0484

2001 0.2766 0.1920 0.0651

To see the yearly trend classified according to firm size in Figure 5,

we find a decline tendency, which is the same as classifying according

to Z-score. However, what different from the results of classifying ac-

cording to Z-score is that the category of “( $0m,$100m)” has a big gap

with categories of “( $100m,$1b)” and “(> $1b)”. In “(> $1b)” cate-

gory, the downward tendency is the most significant one, and for the

analysis based on the Z-score category, the “Z-score>2.99” category

is also the most significant in all of the three groups. That means,

the firms in a healthy financial state or the firms with large asset size

have a more notable decreasing trend in asset correlation comparing

with the other two categories.

Table 19: Means of asset correlation for year 2001-2005 by firm size category for Beta prior

year > $1b ( $100m,$1b) ($0m,$100m)

2005 0.1386 0.1309 0.0181

2004 0.1755 0.1635 0.0383

2003 0.2137 0.1456 0.0291

2002 0.2774 0.2116 0.0474

2001 0.2765 0.1907 0.0645

Figure 5: Means of asset correlation during years 2001-2005 by firm size category

The results of Lopez(2004) in Figure 6 to make a comparision. The

Figure 6: Results of Lopez(2004) for size category

data from year-end 2000 and invested the relationship of asset size

and asset correlation. His results presented that the asset correlation

in the U.S. is between 0.1 and 0.3 basedon asset size categories. Our

results are smaller than the results of Lopez’s in asset size categories

and EDF categories.

5 Conclusion

Asset correlation is the key factor for determining the risk weight in capital requirement of Basel II. There are two findings in Quantita- tive impact study 3 technical guidance (QIS-3) based on the formula for asset correlation in Basel II; negative relationship between asset correlation and default probability, and positive relationship between asset correlation and firm size. Some studies support these findings but some studies do not support them. The problem is that there is no theoretical background for asset correlation formula in Basel II. This study applies Baysian method to the asymptotic single risk factor (ASRF) to investigate the relationship among asset correlation, de- fault probability, and firm size. This approach incorporates the prior information (historical data) and the variability (or precision) of asset correlation into ASRF model. This method provides the theoretical basis for discussing the aforementioned relationship. The data we used in this study include the listed firms in both CRSP and COMPUSTAT databases during years 2001-2005. Equity correlation was used as the proxy for asset correlation since it has been industry practice. Due to the lack of data for probability of default, Alman’s Z-score was used as the proxy of default probability. The relationship was discussed based on the ordered categories of Z-score and firm size, individually.

There are three main findings in this study.

First, the asset correlation is positively correlated with Z-score. This

indicates that the negative relationship between average asset correla-

tion and probability of default. The value of total assets are controlled

by a systematic factor and an idiosyncratic one. The negative rela-

tionship implies that the higher probability of default, the lower the

systematic risk components of a firm. The systematic risk is tied to the general economic environment summarized by the common factor.

Secondly, the study results further indicate that average asset corre- lation is increasing in firm size. That is, as firms increase the book value of their assets, they become more correlated with the general economic environment.

This result is intuitive in the sense that larger firms can generally be viewed as portfolios of smaller firms, and such portfolios would be relatively more sensitive to systematic risks than to idiosyncratic risks.

Finally, the asset correlation is different from year to year, so there may be some other important factors that affect the asset correlation and were not considered in the formula of asset correlation.

There are three limitations of this study. Firstly, we used proxy vari- ables to make comparisons. Secondly, the trend over years should be investigated further to find other important factor for determin- ing asset correlation, third, we use only the firms in the U.S., and as Lopez(2004) mentioned the country effect may be a concern.

References

Altman, E. (1968). Financial ratios, discrimination analysis, and the prediction of corporate bankruptcy. Journal of Banjing and Finance, 23:589–609.

Basel Committee on Banking Supervision (2001a). The internal ratings-based ap- proach: Supporting document to the new basel capital accord. consultative doc- ument. bank for international settlements.

Basel Committee on Banking Supervision (2001b). Quantitative impact study 3

technical guidance.

Basel Committee on Banking Supervision (2002). Potential modification to the committee’s proposals. press release.

Basel Committee on Banking Supervision (2004a). Background note on lgd quan- tification. volume 12, pages 199–232.

Basel Committee on Banking Supervision (2004b). An explanatory note on the basel II IRB risk weight function.

Besag, J. (1974). Spetial interaction and the statistical analysis of life systems.

Journal of Royal Statistical Society, 36:192–236.

Dietsch, M. and J.Petey (2004). Should sme exposures be treated as retail or cor- porate exposures? a comparative analysis of default probabilities and asset corre- lations in french and german smes. Journal of Banjing and Finance, 28:773–788.

D¨ ullmann, K. and Scheule, H. (2003). Determinants of asset correlations of german corporation and implications for regulatory capital. Working paper,University of Regensburg.

Gordy, M. (2003). A risk-factor foundation for risk-based capital rules. Journal of Financial Intemedian, 12:199–232.

Hamerle, A., L. T. and R¨ osch, R. (2003). Credit risk factor modeling and the basel ii irb approach. Discussion Paper, Deutsche Bundesbank.

Hamerle, A., L. T. and Scheule, H. (2004). Forecasting credit portfolio risk. Deutsche Bundesbank, Discussion Paper.

Henneke, J., T. S. (2005). Capital requirement for SMEs under the revised Basel II framework. Working paper,University of Karlsruhe.

Lopez, A. (2004). The empirical relationship between average asset correlation,firm probability of default, adn asset size. Journal of Financial Intermediation, 13:265–

283.

R¨ osch, D. (2002). Correlations and business cycles of credit risk: Evidence from bankruptcies in germany. Working paper,University of Regensburg.

Vasicek, O. (1997). The loan loss distribution. Working paper KMV Corporation.

A Normal and Beta prior

A.1 Normal prior by Z-score category

Table 20: Asset correlation in year 2005 for Normal prior by Z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.0764 0.0035 0.0001 0.0697 0.0764 0.0832 1.81<Z-score<2.99 0.0553 0.0062 0.0001 0.0434 0.0552 0.0677 Z-score<1.81 0.0342 0.0061 0.0002 0.0223 0.0342 0.0462

The sample size of category “Z-score>2.99” is 1422, of category “1.81<Z-score<2.99”

is 454 and of category “Z-score<1.81” is 588.

Table 21: Asset correlation in year 2004 for Normal prior by Z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.0937 0.0035 0.0001 0.0868 0.0937 0.1004 1.81<Z-score<2.99 0.0825 0.0056 0.0001 0.0716 0.0825 0.0935 Z-score<1.81 0.0652 0.0064 0.0001 0.0526 0.0652 0.0779

The sample size of category “Z-score>2.99” is 1499, of category “1.81<Z-score<2.99”

is 520 and of category “Z-score<1.81” is 531.

Table 22: Asset correlation in year 2003 for Normal prior by Z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.0851 0.0041 0.0002 0.0771 0.0850 0.0935 1.81<Z-score<2.99 0.0755 0.0062 0.0002 0.0634 0.0755 0.0874 Z-score<1.81 0.0546 0.0096 0.0003 0.0367 0.0541 0.0746

The sample size of category “Z-score>2.99” is 1339, of category “1.81<Z-score<2.99”

is 490 and of category “Z-score<1.81” is 597.

Table 23: Asset correlation in year 2002 for Normal prior by Z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.1475 0.0050 0.0002 0.1379 0.1474 0.1576 1.81<Z-score<2.99 0.1239 0.0067 0.0002 0.1111 0.1237 0.1374 Z-score<1.81 0.1068 0.0111 0.0005 0.0865 0.1062 0.1300 The sample size of category “Z-score>2.99” is 1112, of category “1.81<Z-score<2.99”

is 516 and of category “Z-score<1.81” is 783.

Table 24: Asset correlation in year 2001 for Normal prior by Z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.1675 0.0065 0.0003 0.1534 0.1654 0.1789 1.81<Z-score<2.99 0.0799 0.0073 0.0002 0.0660 0.0798 0.0944 Z-score<1.81 0.0935 0.0115 0.0004 0.0722 0.0933 0.1171 The sample size of category “Z-score>2.99” is 1209, of category “1.81<Z-score<2.99”

is 480 and of category “Z-score<1.81” is 607.

A.2 Beta prior by Z-score category

Table 25: Asset correlation in year 2005 for Beta prior by Z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.0764 0.0035 0.0001 0.0694 0.0764 0.0832 1.81<Z-score<2.99 0.0546 0.0063 0.0002 0.0422 0.0548 0.0668 Z-score<1.81 0.0341 0.0061 0.0001 0.0216 0.0341 0.0460 The sample size of category “Z-score>2.99” is 1422, of category “1.81<Z-score<2.99”

is 454 and of category “Z-score<1.81” is 588.

Table 26: Asset correlation in year 2004 for Beta prior by Z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.0939 0.0036 0.0001 0.0870 0.0939 0.1009 1.81<Z-score<2.99 0.0824 0.0057 0.0001 0.0712 0.0824 0.0935 Z-score<1.81 0.0648 0.0065 0.0002 0.0522 0.0649 0.0776 The sample size of category “Z-score>2.99” is 1499, of category “1.81<Z-score<2.99”

is 520 and of category “Z-score<1.81” is 531.

Table 27: Asset correlation in year 2003 for Beta prior by Z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.0847 0.0040 0.0001 0.0767 0.0847 0.0924 1.81<Z-score<2.99 0.0754 0.0060 0.0001 0.0637 0.0754 0.0872 Z-score<1.81 0.0532 0.0096 0.0003 0.0347 0.0529 0.0728 The sample size of category “Z-score>2.99” is 1339, of category “1.81<Z-score<2.99”

is 490 and of category “Z-score<1.81” is 597.

Table 28: Asset correlation in year 2002 for Beta prior by Z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.1471 0.0047 0.0002 0.1382 0.1470 0.1563 1.81<Z-score<2.99 0.1238 0.0065 0.0002 0.1113 0.1236 0.1368 Z-score<1.81 0.1057 0.0114 0.0006 0.0844 0.1051 0.1293 The sample size of category “Z-score>2.99” is 1112, of category “1.81<Z-score<2.99”

is 516 and of category “Z-score<1.81” is 783.

Table 29: Asset correlation in year 2001 for Beta prior by Z-score category

category mean sd MC error 2.5% median 97.5%

Z-score>2.99 0.1663 0.0059 0.0004 0.1547 0.1663 0.1783 1.81<Z-score<2.99 0.0795 0.0072 0.0002 0.0655 0.0795 0.0939 Z-score<1.81 0.0927 0.0111 0.0004 0.0721 0.0923 0.1157 The sample size of category “Z-score>2.99” is 1209, of category “1.81<Z-score<2.99”

is 480 and of category “Z-score<1.81” is 607.

A.3 Normal prior by firm size category

Table 30: Asset correlation in year 2005 for Normal prior by firm size category

firm size mean sd MC error 2.5% median 97.5%

> $1b 0.1385 0.004734 1.06E-04 0.1292 0.1386 0.1477 ( $100m,$1b) 0.1309 0.003757 1.04E-04 0.1235 0.1309 0.1382 ( $0m,$100m) 0.01887 0.005328 1.17E-04 0.008544 0.01875 0.0295 The sample size of category “> $1b” is 567, of category “($100m,$1b)” is 1025 and of category “( $0m,$100m)” is 867.

Table 31: Asset correlation in year 2004 for Normal prior by firm size category

firm size mean sd MC error 2.5% median 97.5%

> $1b 0.1757 0.004437 1.08E-04 0.1669 0.1757 0.1842

( $100m,$1b) 0.164 0.003655 1.17E-04 0.1567 0.164 0.171

( $0m,$100m) 0.03871 0.005492 1.40E-04 0.02791 0.03883 0.04918

The sample size of category “> $1b” is 601, of category “( $100m,$1b)” is 1041 and of

category “( $0m,$100m)” is 901.

Table 32: Asset correlation in year 2003 for Normal prior by firm size category

firm size mean sd MC error 2.5% median 97.5%

> $1b 0.2138 0.004531 1.36E-04 0.2049 0.2138 0.2226 ( $100m,$1b) 0.1461 0.004026 1.57E-04 0.1382 0.1462 0.1537 ( $0m,$100m) 0.02973 0.006133 1.67E-04 0.01807 0.02955 0.04229 The sample size of category “> $1b” is 556, of category “( $100m,$1b)” is 989 and of category “( $0m,$100m)” is 874.

Table 33: Asset correlation in year 2002 for Normal prior by firm size category

firm size mean sd MC error 2.5% median 97.5%

> $1b 0.2775 0.005466 3.08E-04 0.2668 0.2775 0.2884 ( $100m,$1b) 0.2111 0.00519 3.42E-04 0.2009 0.2112 0.2209 ( $0m,$100m) 0.04837 0.007916 2.32E-04 0.03363 0.04805 0.06477 The sample size of category “> $1b” is 481, of category “( $100m,$1b)” is 958 and of category “( $0m,$100m)” is 958.

Table 34: Asset correlation in year 2001 for Normal prior by firm size category

firm size mean sd MC error 2.5% median 97.5%

> $1b 0.2766 0.005327 2.13E-04 0.2661 0.2767 0.2871 ( $100m,$1b) 0.192 0.00599 3.94E-04 0.1804 0.1919 0.204 ( $0m,$100m) 0.06512 0.009035 2.66E-04 0.04795 0.06488 0.08344 The sample size of category “> $1b” is 455, of category “( $100m,$1b)” is 934 and of category “( $0m,$100m)” is 895.

A.4 Beta prior by firm size category

Table 35: Asset correlation in year 2005 for Beta prior by firm size category

firm size mean sd MC error 2.5% median 97.5%

> $1b 0.1386 0.004785 1.05E-04 0.1293 0.1386 0.1479 ( $100m,$1b) 0.1309 0.003843 1.36E-04 0.1233 0.1309 0.1383 ( $0m,$100m) 0.01814 0.005514 1.28E-04 0.00742 0.01814 0.02861 The sample size of category “> $1b” is 567, of category “( $100m,$1b)” is 1025 and of category “( $0m,$100m)” is 867.

Table 36: Asset correlation in year 2004 for Beta prior by firm size category

firm size mean sd MC error 2.5% median 97.5%

> $1b 0.1755 0.004359 9.92E-05 0.1671 0.1755 0.184 ( $100m,$1b) 0.1635 0.00378 1.26E-04 0.1562 0.1634 0.1709 ( $0m,$101m) 0.03833 0.005587 1.42E-04 0.0274 0.03842 0.04925 The sample size of category “> $1b” is 601, of category “( $100m,$1b)” is 1041 and of category “($0m,$100m)” is 901.

Table 37: Asset correlation in year 2003 for Beta prior by firm size category

firm size mean sd MC error 2.5% median 97.5%

> $1b 0.2137 0.004598 1.43E-04 0.2047 0.2137 0.2228 ( $100m,$1b) 0.1456 0.004107 1.69E-04 0.1376 0.1456 0.1538 ( $0m,$100m) 0.02905 0.006114 1.48E-04 0.01726 0.02897 0.04125 The sample size of category “> $1b” is 556, of category “( $100m,$1b)” is 989 and of category “( $0m,$100m)” is 874.

Table 38: Asset correlation in year 2002 for Beta prior by firm size category

firm size mean sd MC error 2.5% median 97.5%

> $1b 0.2774 0.005343 2.81E-04 0.2667 0.2773 0.288

( $100m,$1b) 0.2116 0.005021 3.11E-04 0.2017 0.2117 0.2215

( $0m,$100m) 0.04742 0.007732 2.24E-04 0.03258 0.04735 0.06307

The sample size of category “> $1b” is 481, of category “( $100m,$1b)” is 958 and of

category “( $0m,$100m)” is 958.

Table 39: Asset correlation in year 2001 for Beta prior by firm size category

firm size mean sd MC error 2.5% median 97.5%

> $1b 0.2765 0.005602 2.81E-04 0.2656 0.2765 0.2876

( $100m,$1b) 0.1907 0.005668 3.36E-04 0.1794 0.1907 0.2019

( $0m,$100m) 0.06451 0.009267 3.83E-04 0.0473 0.06417 0.08444

The sample size of category “> $1b” is 455, of category “( $100m,$1b)” is 934 and of

category “( $0m,$100m)” is 895.