A Villa

J A Benediktsson

J Chanussot

C Jutten

INDEPENDENT COMPONENT DISCRIMINANT ANALYSIS FOR HYPERSPECTRAL IMAGE CLASSIFICATION

Keywords: Hyperspectral data, Independent Component Analysis, Bayesian Classification

In this paper, the use of Independent Component Discriminant Analysis (ICDA) for remote sensing classification is proposed. ICDA is a non-parametric method for discriminant analysis based on the application of a Bayesian classification rule on a signal composed by independent components. The method is based on the use of Independent Component Analysis (ICA) to choose a transform matrix so that the transformed components are as independent as possible. Then, a non parametric estimation of the density function is computed for each independent component. Finally, the Bayes rule is applied for classification assignment. The obtained results are compared with one of the most used classifier of hyperspectral images (Support Vector Machine) and show the comparative effectiveness of the proposed method.

INTRODUCTION

Hyperspectral images are composed of hundreds of bands with a very high spectral resolution, from the visible to the infra-red region. The wide spectral range, coupled with an always increasing spatial resolution, allows a better characterization of materials and gives the ability to pinpoint ground objects laying on the observed surface, making hyperspectral imagery suitable for land cover classification. On the other hand, the huge amount of high dimensional data increases the computational load and can also degrade the results of the classification process, due to the curse of dimensionality [START_REF] Lee | Analyzing high dimensional multispectral data[END_REF]. Because of this, parametric classifiers such as Maximum Likelihood [START_REF] Duda | Pattern Classification[END_REF] or Bayesian classifier [START_REF] Duda | Pattern Classification[END_REF][START_REF] Besag | Toward Bayesian image analysis[END_REF], based on the assumption that each class can be represented by a multivariate normal model depending on the mean and covariance matrix of the data, lead in general to poor results. More advanced classifiers, like Neural Networks (NN) [START_REF] Haykin | Neural Networks: A Comprehensive Foundation[END_REF] and Support Vector Machines (SVM) [START_REF] Camps-Valls | Kernel-based methods for hyperspectral image classification[END_REF], have been recently proposed for hyperspectral data classification, due to their good ability to handle high dimensional data. Neural Networks suffer basically from two main limitations. First, the number and the size of hidden layers need to be set, and this is not always being a straightforward task. Secondly, a very large number of iterations is sometimes needed to find a solution. Support Vector Machines have even shown better performances than NN, providing good results also in the case of very limited training sets [START_REF] Camps-Valls | Kernel-based methods for hyperspectral image classification[END_REF]. The main limitations of SVM are the training time, quadratically depending on the number of training samples, and the need to find the optimal parameters for the kernel, This work was supported by the Hyper-I-Net project, funded by the European Union usually done using a cross-validation approach. In this work we present a non-parametric method for discriminant analysis based on the application of a Bayesian classification rule on a signal composed by independent components [START_REF] Amato | Independent Component Discriminant Analysis[END_REF]. The main characteristics of the method are the use of Independent Component Analysis (ICA) to retrieve independent components from the original data; non parametric estimation of the density function for each independent component; application of a Bayes rule for classification assignment. The obtained results are compared with one of the most used classifier of hyperspectral images (SVM) and show the comparative effectiveness of the proposed method.

INDEPENDENT COMPONENT DISCRIMINANT ANALYSIS

The proposed method is a generalization of the quadratic discriminant analysis, where the ability of ICA to retrieve independent components is exploited to estimate the marginal densities of the transformed components. These densities, which are hard to estimate when dealing with high dimensional data, can be computed in a much simpler way in an independent space. The risk incurred when performing a classification of a measured vector x into one of K possible classes is given by: R( k|x) =

P K k=1 L(k, k)f k (x)π k P K k=1 f k (x)π k (1)
where π k is the a priori probability that x belongs to the class k, f k is the class-conditional a priori density of k, and L is the cost or loss incurred when assigning the sample x, belonging to the class k, to the class k. In the case of hard classification, this cost is equal to zero if k = k, and is equal to one otherwise.

By choosing k such that the numerator of ( 1) is minimized leads to the so-called Bayes decision rule. In the case of hard classification, the Bayes rule reduces to the following rule: assign x to the class k such that

k = d(x) = argmax{f k (x)π k } k = 1, .., K.
(

The most often applied classification rules are derived by assuming that the class-conditional densities are p-variate normal with mean vectors μ k and covariance matrices Σ k assumed to be non singular. These two parameters are estimated from the training samples according to the following equations:

μk = 1 N k N k X i=1 x ik (3)
978-1-4244-8907-7/10/$26.00 ©2010 IEEE

Σk = 1 N k N k X i=1 (x ik -μk )(x ik -μk ) T (4)
where {x ik , i = 1, . . . , N k } are the training samples of the class k. This approach works well when the class conditional densities are approximately normal and good estimates can be obtained from the training samples. The approach is highly affected by substantial divergence from normal density and by a limited size of training set [START_REF] Friedman | Exploratory projection pursuit[END_REF], as it is often the case for hyperspectral remote sensing data.

The parametric approach to discriminant analysis has been extended to the case where nothing is known about the densities f k but only some assumptions about their general behavior [START_REF] Fix | Discriminatory Analysis -nonparametric discrimination: Consistency properties[END_REF]. Here, the idea is to apply non-parametric density estimators to the training samples and then substitute the obtained estimates in the Bayes decision rule [START_REF] Duda | Pattern Classification[END_REF]. The common used procedure is to use a multivariate kernel density estimator of the form:

fk (x) = K{x -x k ; H k }, ( 5
)
where K denotes a multivariate kernel function and H k is a pdimensional vector of appropriate bandwidths. Multidimensional density estimation is highly affected by the high dimensionality of the data, and pratically is not applicable when the dimension of the data is comparable with the sample size of the classes, such as in the case of hyperspectral data. In these cases, the kernel K is approximated as the product of univariate Gaussian kernel functions, leading to estimates of the form [START_REF] Scott | Multivariate density estimation: theory, practice and visualization[END_REF]:

fk (x) = (2π) -p/2 HN -1 k N k X l=1 p Y j=1 exp n - (xj -x lkj ) 2 2h 2 kj o , (6)
where

H = h k1 • h k1 • • • h kp . ( 7
)
The main drawback of this approach is that some important information for the classification process is not retrieved. When dealing with high-dimensional data where very important information for the classification process can be hidden in relatively low density regions, the tails estimation becomes crucial in order not to degrade the final results. In this case, a Gaussian kernel product estimator can be inappropriate, due to the short tailed normal density. In [START_REF] Amato | Independent Component Discriminant Analysis[END_REF], Amato et al. proposed an interesting approach to circumvent the problems of non-parametric multivariate kernel density estimators. They used ICA to enforce independence to the components of the analyzed data. In the new transformed space the components are as independent as possible. This way, the joint class-conditional density, tricky to estimate in the original data space, can be computed in the independent space in a much simpler way, as the product of marginal class-conditional densities. The results obtained are finally substituted in the Bayes rule for the class assignment. The basic steps of the proposed approach are:

1. Center the data on the k-class, for each class k = 1, . . . , K, and use the ICA to derive the optimal transform Âk according to the training samples of the class. 2. Project the data using the computed transform and use an adaptive univariate kernel density estimator to estimate the density of each component. 3. For a new observation x, the joint density of Y = Âk x is first computed for each class as the product of the estimated marginal densities, since the components are independent. The density of x can be then derived from that of Y with a simple change of variable. The results are then substituted into the Bayes rule to obtain the final assignment.

Later on, we will refer to the above approach as Independent Component Discriminant Analysis (ICDA).

INDEPENDENT COMPONENT ANALYSIS

ICA consists in finding a linear decomposition of observed data into statistically independent components. Given an observation model

x = As, (8)
where x is the vector of the observed signals, A is a scalar matrix of the mixing coefficients and s is the vector of the source signals. ICA finds a separating matrix W such that

y = Wx = WAs, (9)
where y is a vector of independent components. ICA looks for a linear representation that maximizes a non-Gaussianity measure, or minimizes an objective function. A commonly used objective function in ICA algorithms is the mutual information of vector y:

I(y, W) = X i H(yi) -H(y) ( 10
)
where H(yi) and H(y) are the joint and marginal entropy of random variable yi and random vector y, respectively. Many algorithms have been developed in the last decades to estimate independent components. In our experiments, we will use JADE, because of the good performances shown when dealing with hyperspectral data, both for accuracy and computational burden. More details about the general framework of ICA can be found in [START_REF] Comon | Independent component analysis, a new concept?[END_REF][START_REF] Lee | An unifying information-theoretic framework for Independent Component Analysis[END_REF][START_REF] Hyvarinen | Independent Component Analysis[END_REF]. regularly every 10 years, in 1970, 1980-81, 1991 and in 2000. The volcano is located on the South-Western margin of the Eastern volcanic zone in South Iceland. Heklas products are mainly andesitic and basaltic lavas and tephra. AVIRIS data that were collected on a cloud-free day, June 17 1991, were used for the classification. The AVIRIS sensor operates in the wavelengths range from 0.4 μm to 2.4 μm. As on the preavious case, the sensor system has 224 data channels, utilizing four spectrometers, whereas each spectral band is approximately 10nm in width. During the image acquisition, spectrometer 4 was not working properly. This particular spectrometer operates in the wavelength range from 1.84 μm to 2.4 μm (64 bands). These 64 bands were deleted from the imagery along with the first channels for all the other spectrometers, and the remaining 157 data channels were left. The results obtained in our experiments with ICDA have been compared with those obtained by SVM in the full feature space, with a Gaussian kernel and 10 fold cross-validation selection of the kernel's parameter [START_REF] Chang | LIBSVM: A library for support vector machines[END_REF]. The chosen multi class strategy is One versus One. The performances have been computed in terms of Overall Accuracy, that is percentage of samples correctly classified, Average class Accuracy, which represents the average of the classification accuracies for the individual classes, and Kappa coefficient, a parameter that estimates the correct percentage classification without the amount due to chance alone. When applying ICDA, the number of components considered to compute the density estimation has an influence both on the final classification accuracy and on the computational burden. Figure 1 shows the variation of the Kappa coefficient ((a) and (c)) and processing time ((b) and (d)) with respect to the number of independent components retained, for two of the considered data sets, ROSIS and AVIRIS Hekla. Though the number of components has a large influence on the final results, it can be seen that there is a wide region where the proposed method outperforms the SVM. The computational burden of SVM is generally much bigger than the proposed method, and the classification time in the best obtained result in terms of accuracy is, for the AVIRIS Hekla data set, about one half of time for SVM, and for the ROSIS data set, less than one fourth. The reason is that the processing time of the SVM increases quadratically with the size of the training set, and it is longer where a large number of training samples is available (as in the case of ROSIS data set), while ICDA is more affected by the size of the testing samples. The best results are presented in Table 2 and show the effectiveness of the method. In all the three considered data sets the Overall Accuracy and Kappa coefficient are significantly better than in the case of SVM classification.

EXPERIMENTAL RESULTS

Three

CONCLUSIONS

In this work, the use of Independent Component Discriminant Analysis for hyperspectral image classification has been investigated. The proposed approach is based on the application of ICA to the data in order to retrieve independent components, the use of a kernel density estimate to obtain reliable estimation of class dependent densities and the substitution on the Bayes rule for the final assignment. Experiments have been carried out on three different real data sets. These results showed the effectiveness of the method, which provided better results than one of the state-of-the-art hyperspectral classifier, the SVM. Though the classification accuracy is influenced by the number of components retained after applying ICA, this choice is not critical, since there is a large range around the best option for which the proposed method have similar results and outperforms SVM. Moreover, the computational burden of the proposed method is much smaller as compared to SVM. Further developments of this work include a comprehensive research of the influence of the ICA algorithm used to enforce independence. Other techniques, such as FastICA or Infomax, should be investigated.

Fig. 1 .

 1 Fig. 1. Kappa coefficient obtained with SVM (full feature space) and ICDA and classification processing time, with respect to different number of independent components considered, for: (a) and (b) ROSIS, (c) and (d) AVIRIS Hekla data sets. The circle represents the processing time corresponding to the best classification result obtained with ICDA in terms of accuracy.

Table 1 .

 1 Training-test samples for the three considered data sets.

	hyperspectral data sets were considered in this work. The first
	one is an airborne data from the ROSIS-03 with 115 spectral bands
	in the spectral range from 0.43 to 0.86 μm, acquired over the Uni-
	versity of Pavia, Italy. The spatial resolution is 1.3m per pixel. The
	original data set is 610 by 340 pixels. Twelve data channels were re-
	moved due to noise, and the remaining 103 spectral dimensions were
	processed. Training data with nine classes are available. The training
	set is composed by about 10% of all the labeled samples. The second
	data set is a small segment of an AVIRIS (Airborne Visible InfraRed
	Imaging Spectrometer) data set over the agricultural area of Indi-
	ana. The data set is 145 by 145 set of pixels. It is composed of 220
	spectral channels (spaced at about 10 nm) acquired in the 0.4-2.5
	μm region [13]. All 220 bands were processed, without removing
	noisy channels. Sixteen reference classes were considered. Differ-
	ent training sets were randomly constructed from the reference data
	with a total of respectively 320 pixels (20 samples per class). Due to
	the very small size of the training set, each experiment was repeated
	five times to increase the statistical significance of the test and the
	average results are reported. Finally, the third study site is the region
	surrounding the central-volcano Hekla in Iceland, one of the most
	active volcanoes in the country. Since 1970, Hekla has erupted quite
		Training samples Testing samples
	ROSIS University	3921	42776
	AVIRIS Indian Pine	320	10366
	AVIRIS Hekla	600	9636

Table 2 .

 2 Comparison of classification accuracies obtained with the proposed method ICDA (best case) and SVM in the full feature space, in the three analysed data set.

		ROSIS Data Set	AVIRIS Indian Pine	AVIRIS Hekla
	Approach	SVM	ICDA	SVM	ICDA	SVM	ICDA
	OA	81.01% 82.14% 56.00%	66.58% 88.96% 93.91%
	κ	75.86% 77.38% 50.99%	62.50% 86.91% 92.85%
	AA	88.25% 87.48% 70.10%	77.17% 89.44% 92.39%
	Class 1	84.93%	76.40% 84.44%	83.33%	88.36% 96.92%
	Class 2	70.79%	77.74% 36.79%	55.97%	87.25% 95.14%
	Class 3	67.16%	77.42% 40.67%	51.61%	88.24% 94.19%
	Class 4	97.77%	98.07% 72.31%	78.80%	84.94% 96.54%
	Class 5	99.46%	100%	80.40%	84.87%	93.33% 86.11%
	Class 6	92.83%	88.86% 78.93%	92.53%	94.24% 98.56%
	Class 7	90.42%	91.35% 95.38%	96.15%	87.54% 96.06%
	Class 8	92.78%	82.02% 76.11%	90.43%	91.69% 79.54%
	Class 9	98.11%	95.35%	100%	100%	85.88% 87.29%
	Class 10	-	-	53.80%	57.25%	74.20% 80.20%
	Class 11	-	-	39.73%	49.78%	100%	100%
	Class 12	-	-	49.12%	60.07%	97.59% 98.19%
	Class 13	-	-	91.42%	99.15%	-	-
	Class 14	-	-	81.31%	92.30%	-	-
	Class 15	-	-	47.05%	51.89%	-	-
	Class 16	-	-	94.11%	90.53%	-	-

Acknowledgments

The authors would like to acknowledge Prof. Anestis Antoniadis, IMAG Université J. Fourier, for providing the ICDA code and Prof. Paolo Gamba, University of Pavia, for providing the ROSIS data set. This work has been supported by the European Community's Marie Curie Research Training Networks Programme under contract MRTN-CT-2006-035927, HYPER-I-NET.


Keywords: Banking Stocks, Asset Quality, Size and Value Premium JEL Classifications: G10, G12, G15

Banking firms exhibit unique business and financial dynamics that are priced in their stock returns. This paper compares traditional empirical asset pricing models on portfolio of banking firms from fourteen European countries and proposes a banking specific risk factor.

We compared a single factor CAPM with three factors Fama and French model on exchange rate adjusted returns and found substantial support for firm specific factors of size and value.

We propose that asset quality premium (proportion of non-performing loans to total advances and measured as BMG -bad minus good) constitutes an important asset pricing factor for banking stocks. The portfolios sorted on size, value and asset quality explained the maximum variation in returns depicting asset quality as a critical investment factor for banking stocks.

These results have considerable implications for investment appraisals, cost of capital and risk management in financial stocks.

Introduction

Asset pricing has emerged as a dominant theme in financial economics with primary focus on optimal risk return trade-off. The earlier propositions on the subject by Sharpe (1964), [START_REF] Lintner | The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets[END_REF] and [START_REF] Mossin | Equilibrium in a Capital Asset Market[END_REF] revolves around a single factor capital asset pricing model (CAPM) with market risk premium as the only variable that is priced in stock returns.

Despite of some early empirical evidences in favour of CAPM, a large number of academicians suggested additional factors that might affect stock returns. These factors include price earning (P/E) ratio [START_REF] Basu | Investment Performance of Common Stocks in Relation to their Price-Earnings Ratios: A Test of Efficient market Hypothesis[END_REF]], company size [START_REF] Banz | The Relationship between Return and Market Value of Common Stocks[END_REF]] and more importantly book to market and equity size [START_REF] Fama | The Cross-Section of Expected Stock Returns[END_REF]] that triggered alternative explanations for asset pricing puzzle. [START_REF] Fama | The Cross-Section of Expected Stock Returns[END_REF][START_REF] Neuberger | Interest Rate Risk at US Commercial Banks[END_REF][START_REF] Fama | Size and Book-to-Market Factors in Earnings and Returns[END_REF] demonstrated that market risk of CAPM alone is unable to explain the variations in stock returns and firm size (measured as market value of equity) and book to market ratio (value effect) are relevant determinants of expected returns.

The three factor proposition emerged as a significant alternate asset pricing model and related statistical evidences posed some serious challenges to the validity of CAPM. This incited a series of researches to determine if three factor model is robust and holds some theoretical relevance or if such empirical results are synthetic owing to specific data, sample and statistical gimmicks. The exhaustive literature based on US and non US data provide mixed results for three factor model. Some evidences suggest that reports on death of beta are premature and CAPM is still a better predictor of returns, while others propose that size and value premium are deterministic variables for returns and CAPM should be discarded in favour of Fama and French three factor model.

An important sample selection criterion in most of the researches on three factor model is exclusion of financial firms. [START_REF] Fama | The Cross-Section of Expected Stock Returns[END_REF] based their analysis on non financial firms since financial firms inherently are highly leveraged and a similar level of financial leverage in non financial firms would represent severe financial distress. Despite of the significant role of financial firms in economic structure, the studies on their stock returns are limited. [START_REF] Barber | Detecting long-horizon abnormal stock returns: the empirical power and specification of test statistics[END_REF] were the first to examine the cross section of financial stock returns in NYSE vis-à-vis size and value premium and reported significant loadings for three factor model. They compared results with non financial firms and concluded that size and value factors are relevant factors for both financial and non financial firms. [START_REF] Viale | Common Risk Factors in Bank Stocks[END_REF] analysed the risk factors priced in US banking equities. They found no evidence for significance of size and value factors and concluded that market risk premium and shocks to the slope of yield curve are the relevant risk factors.

Financial firms constitute a substantial portion of equity markets both in developed and emerging markets and thus are strong candidates for inclusion in investment portfolios.

Moreover, the regulatory framework of Basel II emphasizes on use of market discipline to analyse the risk taking behaviour of financial firm warrants the need to understand the risk factors relevant to banking equities. Therefore, with unique business and firm characteristics, it would be interesting to observe if conventional asset pricing variables can predict expected returns or if there are more specific factors that could better explain cross section of banking stocks.

We deploy an international portfolio approach on listed banking stocks from fourteen European countries using data that span over 1999 to 2008 and propose an asset quality augmented (bad minus good) Fama and French model. The asset quality 1 of a bank is a unique factor that could affect all aspects of a bank's performance and a substantial decline in asset quality will not only erode the profitability but could ultimately lead to the failure of the bank.

Therefore, banks with better asset quality enjoy better credit standings while banks with low 1 The Asset quality refers to loan quality and both these terms will be used interchangeably.

asset quality, owing to higher non performing loans, have high risk with comparatively higher probability of default.

The empirical evidence is reported on a single factor CAPM, Fama and French three factor model and a four factor asset quality model. The results demonstrate significant factor loadings for all three models. However, the explanatory power of four factor asset quality model was superior to Fama and French three factor model and a single factor CAPM for all sorted portfolios. Therefore; our main findings suggest that a Fama and French model including size and value factors augmented for asset quality best captures the cross section of returns in European banking stocks.

This paper has multiple contributions towards existing literature. Firstly, we extend the asset pricing literature to financial stocks using an exchange rate adjusted international portfolio and a synthetic value weighted index. Secondly, we propose a unique risk factor of banking stocks that is very critical for their intermediary efficiency and long term survival.

Lastly, our results have practical implications for investors and fund managers. The asset pricing of banking stocks for investment and valuation should take into account size, value and asset quality premium. Similarly, for estimating cost of equity of banking firms asset quality should be an integral part. The event studies on banking sector should use a four factor model rather than conventional bank asset pricing variables of interest rates and market premia. Lastly, these results provide an insight into the risk factors of banking stocks that might be of interest for regulators to analyse the risk taking behaviour of such firms.

The rest of the paper is organised as follows. Section II will provide a brief literature review on asset pricing vis-à-vis banking stocks. Section III will discuss the data and methodology. Empirical results are presented in Section IV, while Section V will conclude.

Literature Review

Banking spreads are function of yield on interest generating assets and mark up on interest bearing liabilities and, therefore, are sensitive to interest rate. The dependence of banks' profitability and hence future cash flows on interest rates resulted in literature on financial stocks that proposes use of interest rate variable along with traditional market factor as vital determinant of returns. [START_REF] Stone | Systematic Interest Rate Risk in a Two Index Model of Returns[END_REF] was the first to propose interest rate sensitivity in addition to CAPM market factor to explain variation in stock returns of commercial banks.

Some of the subsequent studies reported empirical results in favour of interest rate factor [START_REF] Fama | Asset returns and inflation[END_REF], [START_REF] Flannery | The Effect of Interest Rate Changes on the Common Stock Returns of Financial Institutions[END_REF]] while others deduced weak or no relationship between interest rate and stock returns [START_REF] Flannery | Asset Pricing, Time-varying Risk Premia and Interest Rate Risk[END_REF]].

However, a caveat to interest rate factor is its variability over time. The factor is reported to have better explanatory power when interest rate volatility is high while in stable interest rate periods; the stock returns remained unaffected from interest rate factor [START_REF] Yourougou | Average Median Average Median Average Median Average Median Average Median Average Median Average Median[END_REF], [START_REF] Choi | The Sensitivity of Bank Stock Returns to Market, Interest and Exchange Rate Risks[END_REF]]. Another set of studies propose that innovations in interest rate variable would have higher explanatory power for returns as compared to expected interest rates. [START_REF] Bae | Interest Rate Changes and Common Stock Returns of Financial Institutions: Revisited[END_REF] studied the impact of expected interest rates and surprises to such expectations for stock returns of depository and non depository institutions. The empirical analysis included three interest rate indices of different maturities and results reported a higher sensitivity for surprises to yield curve with longer maturity. [START_REF] Neuberger | Interest Rate Risk at US Commercial Banks[END_REF] analysed the impact of shocks to yield curve for US commercial banks' returns from 1979 to 1990 and reported a shift in risk sensitivities. The relationship between unexpected changes in interest rates and stock returns was weekend towards the end of the sample period. Dinenis and Staikouras (1996) analysed the impact of expected and innovations in interest rates on UK banks' stocks and found a negative relation between these innovation and returns.

An interesting common point in these studies was observation of higher sensitivity of interest rates towards the start of sample periods that gradually decreases over time. Moreover, none of these studies analysed the traditional asset pricing theories and their ability to explain the variation of returns in banking stocks.

As mentioned earlier, [START_REF] Barber | Detecting long-horizon abnormal stock returns: the empirical power and specification of test statistics[END_REF] studied the pricing of market, size and value factor in US financial stocks. They compared the mean returns for financial and non financial firms sorted on book to market and size factors and found no significant difference.

Similarly, they found significant coefficients on size and value factors for both financial and non financial firms. They concluded that despite of having substantially high financial leverage for financial firms, the market factor along with size and value premium are priced in banking returns and hence such returns could be explained under traditional asset pricing framework. However, they found significant coefficients on excess market returns and shocks to the yield curve under an unconditional ICAPM framework. Similarly, they deduced significant results

for default spread under a conditional ICAPM version. They concluded that only market factors along with innovation in yield curve and default spread are priced in financial returns while book to market and size are not related to variation in returns.

Despite of strong relevance of credit risk as a threat for banking system, the impact on banking stocks is not very clear. Although the interest rates (or its innovations) have been widely discussed as determinant of banking returns yet the impact of credit risk on financial asset pricing has surprisingly been ignored. The informational content of credit risk (disclosure regarding loan loss reserve or non performing loans) provides strong evidence on sensitivity of banking returns to such announcements. [START_REF] Beaver | Financial Reporting, Supplemental Disclosures, and Bank Share Prices[END_REF] studied the relationship between financial reporting, supplementary disclosures and bank share prices for 91 US banks from 1979 to 1983. They found that supplementary variables of interest rate risk and default risk have significant explanatory power. They also deduced that loan loss provisions have incremental significance over traditional valuation variables. They concluded that higher loan loss provisions are associated with higher market values of bank stocks. [START_REF] Griffin | Latin American Lending by Major US banks: The Effects of Disclosures about Nonaccrual Loans and Loan Loss Provisions[END_REF] analysed the impact on stock returns of 13 US banks when they decided to place their Brazilian loans on a non accrual basis and increased loan loss provisions to recognise a higher probability of default. Using event study methodology, they reported that stock market responded adversely to the reclassification of non performing advances and a positive reaction was observed vis-à-vis increase in loan loss reserves. These results depict rational expectations of investors since increase in loan loss provisions is a credible signal about banks intention to sustain their asset quality. They concluded that stock markets discriminate banks on the basis of default risk. [START_REF] Elyasiani | Sensitivity of the Bank Stock Returns Distribution to Changes in the Level and Volatility of Interest Rate: A GARCH-M model[END_REF] used GARCH framework to model interest, market and exchange rate risk for 52 Japanese banking institutions for the period between 1986 and 1996.

The source of these risk estimates was then related with bank specific accounting variables.

These variables included risky debt, risk weighted assets, asset quality, market structure and off balance sheet activities. The results indicated that market and exchange rate risk were significant for all banks while interest rate risk was not relevant throughout sample. The accounting variables as explanatory variables for various risks were significant with negative coefficient on provision for impaired loans. They attributed the negative coefficient to the perception that an increase in these provisions is expected to reduce the credit risk. [START_REF] Ahmed | Bank Loan Loss Provisions: A Reexamination of Capital Management, Earnings Management and Signaling Effects[END_REF] analysed 113 banks with CRSP data over the period 1986 to 1995 and examined the impact of bank loan loss provisions on capital management, earnings management and market signalling. The results validate the capital management hypothesis with diminishing negative relation between loan loss provisions and capital. However, the loan loss provisions had insignificant coefficient with earnings management demonstrating that these provisions have no role in earnings management. For signalling hypothesis, the market adjusted buy and hold returns were regressed on earnings before tax and provisions, loan loss provisions and changes in nonperforming loans and loan charge offs. The results showed a significant negative coefficient on loan loss provisions and change in nonperforming loans while a positive coefficient for earnings were reported. They concluded that a negative sign may refute the signalling hypothesis but significance level demonstrates its relevance in explaining bank stock returns. [START_REF] Hatfield | The Signaling Effects of Bank Loan Loss Reserve Additions[END_REF] examined the market reaction to loan loss reserve additions for 121 announcements between 1980 and 1992 for 33 US bank holding companies.

They used event study methodology and reported a negative reaction to loan loss additions before the announcement. The post announcement reaction was positive and was significantly sustained for several days after the event. They attributed the negative reaction to the surprise factor, as an increase in loan loss provisions indicate erosion in asset quality. However, once the information is known, the investors would perceive the additions as strengthening the risk absorption capacity. They concluded that banking stocks exhibit strong reaction to announcements on default risk and asset quality. [START_REF] Docking | Reaction of Bank Stock Prices to Loan Loss Reserve Announcements[END_REF] analysed the reaction of stock prices to loan loss reserve announcements for 153 banks over a period of 1985 to 1990. They classified the banks into regional and money centred banks and found significant negative relationship between loan loss announcement and abnormal returns. They noted that the loan loss reserves had significant impact despite the fact that these provisions are accounting adjustments and do not represent concurrent cash flow implications. They concluded that bank stock returns are sensitive to loan loss reserves and accounting measure of bank soundness and regulatory pressure are likely to impact the market sentiment for loan loss reserves. [START_REF] Cooper | Evidence of Predictability in the cross-section of Bank Stock Returns[END_REF] investigated the predictability of cross section of bank returns by using prominent variables of financial sector. The sample constituted of 213 banking firms for the period between 1986 and 1999. They used bank specific variables of earnings per share, loans to total assets, loan loss provisions to total loans, non interest income to net income, unused loan commitments to total loans, interest rate swaps to total assets, standby letter of credit to total loans and book value of equity to total assets. Furthermore, they included Fama and French style book to market and size variables to analyse the impact of value and size factors. The results demonstrated that variables of non interest income, loan loss reserves, earnings and leverage were statistically significant while unlike [START_REF] Barber | Detecting long-horizon abnormal stock returns: the empirical power and specification of test statistics[END_REF] they could not deduce a relationship of book to market and size factors with cross section of bank returns. They concluded that bank specific variables have superior predictability power than traditional asset pricing models for cross section of financial returns. [START_REF] Agusman | Accounting and Capital Market Measures of Risk: Evidence from Asian Banks during 1998-2003[END_REF] analysed the relation between accounting and capital market risk measure using a panel data set of 46 banks from ten Asian countries for the period 1998 -2003.They deployed standard market risk variables of total risk, systematic and unsystematic risk and regressed against firm specific variables including standard deviation of return on assets, equity to total assets, gross loans to total assets, loan loss reserve to gross loans and liquid assets to total assets. The results with random effect specification showed that standard deviation of returns and loan loss reserve to gross loans demonstrate a significant positive relation with total risk. Similarly, for non systematic risk, the significant coefficients were reported for loan loss reserve to gross loans and gross loans to total assets. They could not find any significant coefficient when systematic risk was used as independent variable. The pooled OLS exhibit similar results with positive relation of total risk and unsystematic risk with firm variables of standard deviation of return on asset, loan loss reserve to gross loans and gross loans to total assets. However, with pooled OLS, the systematic risk depicted a significant relation with loan loss reserve to gross loans. They concluded that despite of significant differences in banking practices across these Asian countries, the firm specific risks notably credit risk measured as loan loss reserve to gross loans are more relevant than systematic risk and therefore can be used to substitute market based risk measures.

Data and Methodology

Data and Sample

This study focuses on listed banking stocks from 14 European countries including Austria, Belgium, Denmark, Finland, France, Germany, Ireland, Italy, Luxembourg, Netherlands, Spain, Sweden, Switzerland and United Kingdom over the period 1999 -2008.

The primary sample constitutes of banking common stocks from all countries that has daily dividend adjusted price data available on Thomson DataStream. To be included in sample for year t, the selected stocks should have data available on book value of equity, numbers of shares outstanding, non performing loans and gross advances for year t -1. We compute firm size as market value of equity (price times number of shares outstanding), book to market ratio and an asset quality measure of non performing loans to gross advances (NPL/GA).

Furthermore, following [START_REF] Fama | Common Risk Factors in the Returns on Stocks and Bonds[END_REF], we address the survivorship bias by including only those stocks which are listed for atleast two years. Similarly, the stocks classified as Dead or Delisted are discarded from the sample. Lastly, we account for the non synchronous trading phenomenon in the final sample as in presence of sleeping stocks the estimated risk parameters are likely to be biased. Therefore, in final sample, we consider only those banking stocks which demonstrate active daily trading and exclude stocks that have at least 85% non-zero returns in one year. Based on this criterion, the number of selected banking stocks from each country for various years is reported in Table 1.

[Insert Table 1 about here]

The sample consists of 113 baking firms in 1999 that increased to 233 in 2008. We observed that the variation in sample is because of the thin trading criteria as most of the banks had no or substantially inactive trading during the early years. However, over time with increase in overall market activity, the volumes for banking firms increased as well and our sample is almost doubled in 2008. Since all of the stocks in the sample exhibit active returns our sample is free from any bias that could arise due to non synchronous trading. On average, the sample contains more banks from Denmark, Germany, France, Switzerland and United Kingdom while contribution of Austria, Belgium and Luxembourg remained low.

The selected stocks from each country will be combined to form international portfolios. An important consideration in formation of such portfolios is common currency as we cannot directly compare risk and returns emanating from stocks denominated in different currencies [START_REF] Fletcher | On the Conditional Relationship between Beta and Return in International Stock Markets[END_REF]]. Most of the countries included in the sample have prices and other data in Euros; however, for Denmark, Sweden, Switzerland and United Kingdom, the daily prices are denominated in local currencies. To mitigate the impact of exchange rate, we convert the prices and fundamental data of these four countries into common currency (Euro) at spot rate. Once the prices are homogenized to Euro, the individual daily logarithmic returns would be computed. These would take the form , where R i(t) represents return for stock i on day t while P i(t) and P i(t-1)

represent prices (in euro) for stock i on day t and t -1 respectively. These individual returns are then used to compute value weighted portfolio returns sorted for size, book to market and asset quality. Table 2 present the yearly statistics of bank size in our sample for all countries.

[Insert Table 2 about here]

The average size statistics for all countries are following an increasing trend from 1999 to 2008. The banks from United Kingdom exhibit larger average size while Denmark, Netherlands and Switzerland have more banks that are smaller in size. Table 3 shows country wise averages and median of book to market ratio.

[Insert Table 3 about here]

The average book to market ratio is on a higher side for Swiss banks while the ratio remained low for United Kingdom. This represents more value stocks among Swiss banks and higher number of growth stocks in United Kingdom. Considering the utmost relevance of equity as risk absorption cushion in financial sector, Swiss banks with average small size are justified to have an average value rating with respect to book to market ratio. Similarly, United Kingdom with relatively larger banks has the tendency to contribute more growth stocks.

The statistics for non performing loans to gross advances are included in Table 4. The obvious fact from these numbers is a significant increase in ratio for almost all countries in 2008 as compared to 2007. This is clearly attributable to the financial crises in the banking system as a significant increase in classified loans is among the main causes to trigger the financial panic in 2008. The average NPL/GA for France and Germany during 1999 to 2008 remained higher than other countries depicting bad asset quality for French and German banks, while Danish banks showed better asset quality with low NPL/GA ratio.

[Insert Table 4 about here]

Portfolio Construction

In order to capture the impact of traditional size and value premium along with the augmented factor of asset quality the portfolios are constructed on a three way sort. We combined banking stocks from all countries selected as per sample criterion. Then using median of market value of equity for year t, we classified all stocks into two size portfolios as big (B) and small (S) for year t+1. The next step is creating book to market portfolios in two size groups based on book to market value of all stocks with top 30% as high (H), middle 40% as medium (M) and lower 30% as low (L). This results in a total of six market value and book to market portfolios with three book to market portfolios in each size group.

Furthermore, we classified stocks in each of these six portfolios according to their asset quality in two stock groups. There are two main ratios that are widely accepted as possible indicators of asset quality and credit risk. These include non performing loans to gross advances (NPL/GA) and loan loss reserve to gross loans [START_REF] Agusman | Accounting and Capital Market Measures of Risk: Evidence from Asian Banks during 1998-2003[END_REF]].

However, to estimate asset quality, NPL/GA is preferred over loan loss reserve to gross loans because loan loss reserve is directly dependent on a bank profitability and banks with higher pre provision earnings could create higher provisions and vice versa. On the contrary, NPL/GA directly assesses the bank's capacity in its core business as this ratio is dependent on prudent and clean lending system and credit risk controls. A higher ratio of NPL/GA indicates higher proportion of non-performing loans vis-à-vis total loans and ultimately requires more provisioning. Therefore, asset quality could affect a bank's performance by impacting both revenues and expenses. Initially, bad loans reduce the interest revenues on loan portfolios and later a provision is required to expense out the impact of such loans. To include asset quality factor in our proposition, the stocks having a NPL/GA2 value higher than median are considered as bad quality banks (B d ), while those with a lower value are classified as stocks having good asset quality (G d ). In this manner we obtain a total of twelve value weighted portfolios that contains stocks which are sorted for size, book to market and asset quality.

These portfolios are rebalanced every year in June for the sample period based on size, book to market and asset quality. The portfolio construction procedure is illustrated in figure 1.

[Insert Figure 1 about here]

The twelve portfolios are titled as per their size, book to market value and asset quality. The portfolio BHB d will represent stocks that are big in size with high book to market ratio and bad asset quality. Similarly SLG d will represent stocks that are small in size with low book to market and good asset quality.

Variables and Model Estimation

In this study we will examine the traditional asset pricing propositions for our international banking stock portfolios. These include a single factor CAPM, a general three factor Fama and French model and a four factor model augmented for asset quality. The testable versions of three models that will be analysed for each of the twelve portfolios are mathematically represented as follows.

where asset quality premiums respectively. The construction of these variables is explained below.

Excess Portfolio Returns

The excess returns on each of twelve portfolios is the difference between the value weighted portfolio intraday returns and risk free rate. Since Euro is our reference currency, we will use daily effective rate of Euro Overnight Index Average (EONIA) as risk free proxy from 1999 to 2008. EONIA is a reference rate used as benchmark for European money and capital markets and quoted by European central bank.

Market Premium

The market premium is the excess return of market index over risk free EONIA overnight index. As we are using an international portfolio approach we construct a value weighted international index using stock indices of all fourteen European countries and respective market capitalization. An alternate could be to use some MSCI Europe benchmark; however, many banking stocks are not part of such indices. Moreover, it might not be a true representative of the risk return environment in which these banking stocks are traded.

Therefore, construction of a value weighted international index from all local indices using their market capitalization is logical. The logarithmic returns for individual country index is calculated and then combined to formulate returns on an international index. Mathematically, the intraday market returns can be represented as where represents market returns at day t for our international index, indicates the weight of local market index3 vis-à-vis market capitalization and is the daily return on local market index for day t.

Size Premium

The size premium captures the impact of firm size on their respective returns. The intuition behind size premium is simple. Small firms often depict lower financial flexibility and are highly sensitive to business risk conditions as compared to large firms. This result in augmentation of risk for small firms and investors are likely to demand a premium to invest in firms that are relatively smaller in size. In banking stocks, due to inherent business dynamics, the size impact could be even more critical. The equity offers risk absorption capacity against various banking risks and that is primary reason why regulators are sensitive about capital requirements in banking firms. The SMB factor is the difference in average returns of six value weighted portfolios of banking firms with small market capitalization and six portfolios of stocks with big capitalization and hence will be neutral of value and asset quality premiums. A positive SMB would reflect higher returns for small firms over big firms and vice versa. Mathematically, size premium is calculated as SMB

Value Premium

The HML variable accounts for value premium in stock returns. The factor is relevant as firms with high book to market ratio demonstrated higher returns vis-à-vis firms with low book to market ratio. The stocks with high book to market ratio are called value stocks while those with low book to market ratios were termed growth stocks. [START_REF] Fama | The Cross-Section of Expected Stock Returns[END_REF] proposed that value premium is significantly priced in stock returns along with market risk and size premium. The value premium attributes higher returns for value stocks over growth stocks. The value stocks having high book to market ratio are suspected to be firms with distressed characteristics and it would be rational to demand a premium for investment in such stocks. The HML factor is neutral with respect to size and asset quality. This is estimated as the difference between the average returns on four value weighted portfolios with high book to market ratio and four portfolios with low book to market ratio. A positive HML would reflect higher returns for value stocks compared to that of growth stocks and vice versa. The value premium takes the form HML =

Asset Quality Premium

We propose a bank specific risk factor that would add explanatory power to cross section of returns. As mentioned earlier, asset quality is basic determinant of a bank performance and a vital aspect in credit ratings of a financial institution. The management of asset quality includes the prudence and procedures required by the banks to extend credit.

This also includes effective monitoring systems and limits to control credit risk. A sustained asset quality is required for long term survival of a financial institution and banks with low asset quality are likely to face credit and liquidity crunch. Therefore, investors are likely to demand a premium for investing in stocks that have bad asset quality. We measure asset quality by ratio of NPL/GA. A higher ratio will reflect higher proportion of bad loans in total loans and such banks would be considered as ones having a bad asset quality and vice versa.

The asset quality premium B d MG d is defined as average returns on six value weighted portfolios having bad asset quality and six portfolios that depict good asset quality. A positive factor would demonstrate premium returns for firm with bad asset quality as compared to banks having good asset quality. Asset quality factor is sorted on NPL/GA ratio and neutral of size and value factors. The asset quality premium is represented as

B d MG d =
The daily descriptive statistics for twelve mimicking portfolios of size, value and asset quality along with returns on our synthetic index and risk free rate are reported in table 5. For all portfolios, the volatility in returns is increasing over time with maximum standard deviation in 2008 which was a bad year for banking sector. The erosion in fundamentals of banking sector was reflected in their stock returns with average negative returns for the crisis year for all of the twelve portfolios.

[Insert Table 5 about here]

Empirical Results

The regression results for single factor CAPM, three factor Fama and French and asset quality augmented four factors model on size, value and asset quality mimicking portfolios are reported in Table 6, 7 and 8. The single factor CAPM is significant for all twelve portfolios with market risk premium explaining the variation in returns. The maximum explanatory power is for big banks with low book to market and bad asset quality (BLB d )

with an R 2 of 78.9%. The minimum explanatory power is demonstrated for small banks with high book to market and good asset quality (SHG d ) with an R 2 of 3.9%. The market risk premium is positive for all portfolios while intercept is insignificant in all instances indicating that CAPM is able to capture some of the variations for all of our sorted portfolios.

[Insert Table 6 about here]

The incremental explanatory power, for all twelve portfolios, has increased for Fama

[Insert Table 7 about here]

The results for four factor model augmented by asset quality are even more encouraging with maximum adjusted R 2 of 85% for BLB d portfolio and a minimum of 13.5%

for SMG d . For all twelve portfolios, we observe a substantial increase in explanatory power with increase in adjusted coefficient of variation. All intercepts were insignificant reflecting a valid specification for expected returns. The coefficients on all four risk premia were significant and we observe a significant value coefficient for BMB d which was insignificant in

[Insert Table 8 about here]

It is notable that for all three time series models the R 2 increases from small banks to large banks. This was expected because the large banks are more actively traded than small banks and market wide factors are more likely to influence the active stocks. The inclusion of asset quality premium increase the explanatory power of all portfolios, however, for small size banks it is interesting to note that the increase is phenomenal and we observe a significant increase in R 2 vis-à-vis big banks. This implies that although asset quality is vital for all banks, an increase in infected portfolio has devastating impact for small banks. This is logical as banks with smaller equity base would find it difficult to absorb the loan losses emanating from a deteriorating asset quality. Lastly, as mentioned earlier small less active stocks are more likely to be influenced by firm specific factors and asset quality is the most critical of the bank specific risk factors. . These results clearly propose that in case of banking stocks, the variation in returns is jointly explained by market, size, value and a banking unique factor of asset quality.

Conclusion

This paper attempts to explain the behaviour of exchange rate adjusted returns of banking stocks from fourteen European countries under traditional asset pricing framework of CAPM and Fama and French three factors model. Given the unique business and financial risk of banking stocks we propose to augment the traditional size and value factors model by including an asset quality premium. The selected banking stocks were sorted into twelve portfolios at the interaction of size, value and asset quality factors. Our empirical results provide evidence that augmented four factor model explain maximum variation in stock returns of banking stocks. Moreover, small banks with their limited scale are more sensitive to deterioration in asset quality with limited cushion to absorb loan losses.

These results have policy implications for investors in specific and regulators in general. The investment in financial stocks should take into account the asset quality of the credit portfolio of these institutions. Banks with higher infection are likely to face pressure on their spreads with reduced interest revenues. Simultaneously, the loan loss provisions will increase the non interest expense affecting the overall profitability of the bank and ultimately the impact will be absorbed in equity thus reducing the risk absorption capacity. Since, this treatment of credit infection is unique to banking firms, the investment in financial stocks warrants the pricing of asset quality.

  three factor model. The signs of the coefficients are also consistent with presence of asset quality premium with positive B d MG d factor for banks with bad asset quality (BHB d , BMB d , BLB d , SHB d , SMB d , SLB d ) and negative with good asset quality (BHG d BMG d BLG d SHG d SMG d SLG d ).

  and French three factor model indicating that additional variation is explained by incorporating size and value premium. The adjusted R 2 for BLB d portfolio increased to approximately 82% while that of SHG d increased to 16.9%. The coefficients of market premium, size and value factors are significant for all portfolios except big banks with medium book to market and bad asset quality (BMB d ) where we could not find evidence for value premium. The signs of coefficients are consistent with size and value fallacy with positive SMB coefficient for small stocks (SHB d , SHG d , SMB d , SMG d , SLB d , SLG d ) and negative for big stocks (BHB d , BHG d , BMB d , BMG d , BLB d , BLG d ). Similarly, we report negative HML coefficients for low book to market stocks (SLB d , SLG d , BLB d , BLG d ) and a positive value factor for high book to market stocks (BHB d , BHG d , SHB d , SHG

d

). These results clearly indicate that size and value premia are priced in banking stocks in addition to the market risk premium.

Table 1 : Country Wise Statistics (No of Firms) 1999 -2008

 1 Moreover, with deregulation in financial sector, the regulators under Basle II framework focus on market discipline to evaluate and control risk taking behaviour of banking firms. To implement such market based measures, it is critical to understand the risk factors that are priced in banking equities and among all such factors we propose the fundamental relevance of asset quality premium.

	Figure 1: Portfolio Construction Procedure
	Market Capitalization	Book to Market	Asset Quality	Portfolios
			Bad Asset Quality	BHB d
		High B/M
			Good Asset Quality	BHG d
			Bad Asset Quality	BMB d
	Big MV	Medium B/M
			Good Asset Quality	BMG d
			Bad Asset Quality	B/LB d
		Low B/M
			Good Asset Quality	BLG d
			Bad Asset Quality	SHB d
		High B/M
			Good Asset Quality	SHG d
			Bad Asset Quality	SMB d
	Small MV	Medium B/M
			Good Asset Quality	SMG d
			Bad Asset Quality	SLB d
		Low B/M
			Good Asset Quality	SLG d

Table 3 : Country Wise Firm Statistics (Book to Market )

 3

	1999	2000	2001	2002	2003	2004	2005	2006	2007	2008

Average Median Average Median Average Median Average Median Average Median Average Median Average Median Average Median Average Median Average Median

	Austria	0.87	0.64	1.29	0.80	1.26	0.71	1.24	0.71	1.38	0.77	1.55	0.72	1.24	0.90	1.30	0.93	1.26	0.77	1.33	0.85
	Belgium	2.36	0.42	3.43	0.42	2.83	0.51	1.87	0.69	1.43	0.93	1.51	0.85	1.36	0.71	1.37	0.60	1.33	0.53	1.95	1.22
	Denmark	1.27	1.27	1.34	1.29	1.22	1.16	1.23	1.20	1.01	1.00	0.82	0.83	0.66	0.67	0.58	0.61	0.53	0.56	0.89	0.90
	Finland	0.88	0.88	0.85	0.85	0.79	0.79	0.66	0.66	0.84	0.84	0.68	0.68	0.68	0.68	0.61	0.61	0.53	0.53	0.61	0.61
	France	1.25	1.15	1.15	1.11	0.92	0.95	1.01	0.95	1.24	1.28	1.11	1.22	1.37	1.41	1.40	1.49	1.33	1.64	2.36	2.78
	Germany	0.75	0.69	0.55	0.52	0.61	0.74	0.99	1.00	1.36	1.25	0.98	0.97	0.76	0.68	0.48	0.56	0.46	0.47	0.65	0.59
	Ireland	0.36	0.37	0.51	0.50	0.38	0.35	0.34	0.33	0.41	0.42	0.42	0.46	0.40	0.42	0.44	0.51	0.45	0.44	1.07	1.14
	Italy	0.57	0.54	0.66	0.60	0.69	0.62	0.81	0.66	0.79	0.76	0.74	0.69	0.72	0.76	0.67	0.63	0.68	0.65	1.10	1.10
	Luxembourg	0.65	0.65	0.21	0.21	0.10	0.10	0.06	0.06	0.12	0.12	0.16	0.16	0.27	0.27	0.54	0.54	0.48	0.48	0.96	0.96
	Netherlands	0.36	0.36	0.37	0.37	0.44	0.44	0.48	0.48	0.61	0.61	0.62	0.62	0.61	0.61	0.45	0.45	0.55	0.55	0.60	0.60
	Spain	0.39	0.40	0.46	0.45	0.50	0.49	0.49	0.49	0.52	0.53	0.49	0.49	0.49	0.48	0.44	0.46	0.46	0.43	0.71	0.69
	Sweden	0.51	0.51	0.51	0.51	0.55	0.58	0.64	0.65	0.80	0.79	0.67	0.65	0.64	0.64	0.59	0.60	0.56	0.55	0.93	0.92
	Switzerland	1.24	0.84	2.08	0.85	1.70	0.88	1.85	0.73	2.30	0.90	0.79	0.78	5.02	0.74	4.75	0.60	4.68	0.55	4.78	0.66
	United Kingdom	0.27	0.29	0.46	0.44	0.40	0.40	0.42	0.44	0.49	0.50	0.51	0.52	0.48	0.52	0.55	0.50	0.62	0.57	1.19	0.94

Table 4 : Country Wise Firm Statistics Firm (Non Performing Advances to Gross Finances)

 4

	Country	1999	2000	2001	2002	2003	2004	2005	2006	2007	2008

Average Median Average Median Average Median Average Median Average Median Average Median Average Median Average Median Average Median Average Median

	Austria	2.20	1.91	2.40	1.85	3.10	3.13	2.79	1.91	2.86	0.97	2.32	2.07	2.21	1.78	2.13	2.24	2.12	1.17	3.93	3.28
	Belgium	2.35	2.50	2.80	2.12	2.90	1.97	3.00	1.98	2.60	2.10	2.61	2.49	2.67	2.17	1.89	1.65	1.90	1.63	1.53	0.95
	Denmark	0.70	0.35	0.90	0.50	4.39	0.92	1.50	1.07	1.42	1.06	1.36	1.07	1.40	1.00	0.96	0.62	0.65	0.61	2.46	0.92
	Finland	0.60	0.50	0.60	0.50	1.31	0.41	1.93	0.44	1.42	0.41	1.44	0.45	3.78	0.65	1.94	0.55	0.54	0.50	0.95	0.94
	France	5.12	4.97	5.65	5.12	6.72	5.21	6.51	4.82	7.14	6.77	6.22	5.10	4.39	3.87	4.27	3.49	4.19	3.36	4.40	3.21
	Germany	4.70	3.35	4.60	3.20	3.46	3.13	4.85	4.30	4.19	4.15	4.99	3.71	6.30	3.33	3.33	2.43	3.16	2.16	2.94	2.15
	Ireland	1.00	0.37	0.97	0.40	0.92	0.90	0.91	0.91	0.76	0.74	0.72	0.62	0.97	0.78	0.89	0.87	0.77	0.76	2.32	1.31
	Italy	2.51	1.87	2.35	1.59	2.39	1.60	5.99	1.78	1.95	2.20	10.48	3.19	4.41	4.02	4.45	3.58	4.85	3.45	5.93	4.15
	Luxembourg	2.85	2.11	2.97	2.53	3.15	3.31	3.37	3.80	2.74	2.80	2.48	2.36	2.28	1.81	1.49	1.28	3.86	0.46	3.34	1.66
	Netherlands	1.80	1.35	2.33	1.73	1.77	1.20	3.77	1.89	1.48	1.65	1.28	1.37	1.51	1.51	1.85	1.43	1.40	1.12	1.94	1.79
	Spain	1.20	0.98	1.90	0.73	4.32	2.70	3.75	1.90	3.93	3.72	1.49	0.95	1.21	0.80	1.06	0.67	1.88	0.93	3.70	2.98
	Sweden	1.60	1.65	1.50	1.92	2.37	1.08	2.73	0.72	3.26	0.76	2.74	0.51	2.94	1.09	3.09	0.92	1.86	0.83	1.60	1.25
	Switzerland	4.10	3.97	2.30	1.97	1.86	2.36	1.95	1.41	3.31	1.83	3.11	1.88	2.71	1.73	2.47	1.55	2.19	1.32	1.58	1.14
	United Kingdom	2.50	1.27	2.60	1.90	3.12	2.51	6.88	3.28	8.18	2.70	3.73	1.71	2.21	0.86	2.45	1.16	2.76	1.17	3.42	2.44

Table 5 : Descriptive Statistics of Returns (1999 -2008)

 5

	Year BHB d	BHG d	BMB d	BMG d	BLB d	BLG d	SHB d	SHG d	SMB d	SMG d	SLB d	SLG d	Index Risk Free
							Mean
	1999 0.029%	0.032%	0.052%	0.035%	0.044% -0.003% 0.036%	0.005%	0.016%	0.008%	0.068%	0.046%	0.076%	0.011%
	2000 0.000%	0.083%	0.035% -0.012% 0.039%	0.055%	0.056%	0.037%	0.020%	0.001%	0.027%	0.011% -0.030%	0.016%
	2001 -0.063% 0.011% -0.042% 0.040% -0.028% -0.048% 0.008%	0.049% -0.042% 0.016% -0.011% -0.082% -0.054%	0.017%
	2002 -0.022% 0.032% -0.034% 0.010% -0.070% -0.065% -0.034% 0.041%	0.003%	0.022% -0.191% -0.040% -0.125%	0.013%
	2003 0.085%	0.094%	0.089%	0.141%	0.059%	0.101%	0.042%	0.142%	0.074%	0.130%	0.062%	0.016%	0.060%	0.009%
	2004 0.046%	0.152%	0.049%	0.075%	0.050%	0.110%	0.058%	0.071%	0.049%	0.067%	0.033%	0.048%	0.053%	0.008%
	2005 0.083% -0.047% 0.092%	0.108%	0.068%	0.164%	0.036%	0.042%	0.080%	0.104%	0.003%	0.099%	0.069%	0.008%
	2006 0.057%	0.041%	0.086%	0.058%	0.096%	0.147%	0.025%	0.053%	0.024%	0.087%	0.009%	0.064%	0.066%	0.011%
	2007 -0.055% -0.025% -0.081% -0.023% -0.047% -0.055% -0.069% -0.016% -0.024% -0.053% -0.120% -0.011% -0.012%	0.015%
	2008 -0.337% -0.165% -0.198% -0.117% -0.200% -0.123% -0.270% -0.194% -0.292% -0.213% -0.146% -0.085% -0.155%	0.016%
							Median
	1999 0.001%	0.002%	0.033%	0.000%	0.002% -0.007% 0.026% -0.002% 0.036%	0.023%	0.128% -0.008% 0.115%	0.010%
	2000 -0.001% 0.049%	0.043% -0.004% 0.119%	0.014%	0.008%	0.035% -0.003% 0.040%	0.022%	0.023%	0.012%	0.017%
	2001 -0.034% 0.031%	0.021%	0.071%	0.062%	0.051%	0.012%	0.048%	0.001%	0.003%	0.013% -0.019% -0.011%	0.017%
	2002 0.001%	0.042%	0.015%	0.023% -0.044% 0.000% -0.062% 0.023%	0.007%	0.022% -0.070% -0.057% -0.136%	0.013%
	2003 0.016%	0.049%	0.108%	0.058%	0.042%	0.053%	0.041%	0.087%	0.096%	0.088%	0.001%	0.003%	0.045%	0.008%
	2004 0.043%	0.063%	0.062%	0.071%	0.073%	0.080%	0.054%	0.051%	0.059%	0.032%	0.020%	0.020%	0.071%	0.008%
	2005 0.099% -0.013% 0.063%	0.151%	0.061%	0.066%	0.026%	0.040%	0.000%	0.098%	0.030%	0.045%	0.086%	0.008%
	2006 0.123%	0.064%	0.176%	0.115%	0.127%	0.180%	0.015%	0.040%	0.029%	0.112% -0.010% 0.066%	0.181%	0.011%
	2007 -0.008% 0.000% -0.036% -0.003% 0.017%	0.025% -0.055% -0.007% 0.000% -0.013% -0.134% 0.000%	0.085%	0.015%
	2008 -0.406% -0.054% -0.280% -0.094% -0.223% -0.055% -0.177% -0.137% -0.338% -0.114% -0.114% -0.037% -0.110%	0.016%
						Standard Deviation
	1999 0.579%	1.214%	0.792%	0.735%	0.991%	1.028%	0.259%	0.227%	0.471%	0.257%	1.218%	1.421%	0.889%	0.001%
	2000 0.673%	0.716%	0.549%	0.945%	0.792%	1.082%	0.689%	0.283%	0.619%	0.306%	1.438%	0.941%	1.006%	0.002%
	2001 1.100%	0.772%	0.816%	0.859%	1.092%	1.162%	1.345%	0.292%	1.033%	0.423%	2.204%	0.777%	1.196%	0.002%
	2002 1.586%	0.819%	0.932%	0.629%	1.407%	1.305%	0.743%	0.207%	0.876%	0.660%	1.726%	0.595%	1.489%	0.001%
	2003 0.891%	0.671%	0.888%	0.842%	0.748%	1.015%	0.494%	0.436%	1.346%	0.361%	1.867%	0.500%	1.051%	0.001%
	2004 0.457%	1.033%	0.481%	0.700%	0.458%	0.638%	0.465%	0.336%	0.660%	0.342%	0.484%	0.526%	0.612%	0.001%
	2005 0.431%	0.982%	0.473%	0.531%	0.485%	1.081%	0.257%	0.407%	0.746%	0.330%	0.000%	0.708%	0.514%	0.001%
	2006 0.603%	0.851%	0.813%	1.060%	0.726%	1.115%	0.419%	0.485%	0.819%	0.406%	0.638%	0.751%	0.780%	0.001%
	2007 0.829%	1.137%	1.077%	0.872%	1.018%	0.855%	0.432%	0.458%	0.702%	0.593%	1.371%	0.531%	0.976%	0.001%
	2008 2.516%	1.335%	1.686%	1.329%	1.766%	0.914%	0.973%	0.833%	2.163%	0.694%	2.107%	0.520%	1.520%	0.001%

Table 6 : Single Factor (CAPM) Regression on Portfolios Sorted for Size, Book to Market and Asset Quality

 6

		α	β 1	t(α)	t(β 1 )	R 2
	BHB d	-0.0001	0.588	-0.696	34.965	0.325
	BHG d	0.0002	0.278	0.959	15.991	0.091
	BMB d	0.0001	0.723	1.018	77.584	0.703
	BMG d	0.0003	0.352	1.839	23.425	0.178
	BLB d	0.0001	0.862	0.596	97.374	0.789
	BLG d	0.0003	0.314	1.351	16.942	0.102
	SHB d	-0.0002	0.169	-1.215	13.282	0.065
	SHG d	0.0001	0.082	1.452	10.127	0.039
	SMB d	-0.0002	0.317	-1.064	16.732	0.099
	SMG d	0.0001	0.090	0.958	10.348	0.040
	SLB d	-0.0002	0.857	-0.855	39.394	0.379
	SLG d	0.0000	0.176	-0.030	12.361	0.057

Table 7 : Three Factor Regression on Portfolios Sorted for Size, Book to Market and Asset Quality

 7

		α	β 1	β 2	β 3	t(α)	t(β 1 )	t(β 2 )	t(β 3 )	Adj R 2
	BHB d	-0.0002	0.583	-0.548	0.458	-1.372	30.247	-16.119	16.832	0.477
	BHG d	0.0001	0.300	-0.500	0.515	0.589	15.035	-14.203	18.303	0.293
	BMB d	0.0000	0.629	-0.371	-0.021	0.323	55.375	-18.536	-1.281	0.740
	BMG d	0.0002	0.198	-0.677	0.023	1.138	11.060	-21.475	0.911	0.315
	BLB d	0.0000	0.731	-0.345	-0.179	-0.160	68.052	-18.199	-11.773	0.816
	BLG d	0.0000	-0.185	-1.083	-0.884	0.246	-10.025	-33.213	-33.904	0.472
	SHB d	-0.0001	0.395	0.404	0.476	-0.616	26.614	15.447	22.748	0.247
	SHG d	0.0002	0.199	0.197	0.258	2.112	20.381	11.431	18.741	0.169
	SMB d	-0.0001	0.567	0.721	0.288	-0.368	24.428	17.614	8.796	0.203
	SMG d	0.0001	0.163	0.172	0.119	1.371	14.723	8.795	7.585	0.079
	SLB d	-0.0001	0.790	0.730	-0.879	-0.450	35.039	18.341	-27.616	0.607
	SLG d	0.0000	0.140	0.251	-0.350	0.238	8.071	8.199	-14.257	0.173

Table 8 : Four Factor Regression on Portfolios Sorted for Size, Book to Market and Asset Quality

 8

		α	β 1	β 2	β 3	β 4	t(α)	t(β 1 )	t(β 2 )	t(β 3 )	t(β 4 ) Adj R 2
	BHB d	0.0000	0.088	-0.931	0.378	1.031	-0.275	4.287	-31.742	17.152	36.887	0.660
	BHG d	0.0000	0.575	-0.286	0.559	-0.574	-0.031	23.034	-8.028	20.859	-16.886	0.365
	BMB d 0.0001	0.397	-0.551	-0.058	0.483	1.400	29.974	-29.133	-4.068	26.823	0.797
	BMG d 0.0001	0.266	-0.624	0.034	-0.142	0.970	11.312	-18.583	1.345	-4.449	0.320
	BLB d	0.0001	0.530	-0.500	-0.211	0.418	0.747	41.396	-27.333	-15.318	23.972	0.850
	BLG d	0.0000	0.014	-0.929	-0.852	-0.415	-0.244	0.582	-27.493	-33.561	-12.884	0.505
	SHB d	0.0000	0.322	0.348	0.465	0.152	-0.399	16.542	12.512	22.220	5.717	0.256
	SHG d	0.0001	0.292	0.269	0.273	-0.193	1.728	23.177	14.945	20.211	-11.276	0.208
	SMB d	0.0000	0.242	0.470	0.236	0.677	0.273	8.338	11.328	7.569	17.140	0.285
	SMG d 0.0001	0.281	0.264	0.138	-0.246	0.922	19.813	12.991	9.029	-12.733	0.135
	SLB d	0.0001	0.291	0.343	-0.960	1.041	0.629	11.340	9.367	-34.857	29.840	0.709
	SLG d -0.0001	0.442	0.485	-0.301	-0.629	-0.588	21.010	16.139	-13.328	-21.960	0.305

NPL/GA is the value of non-performing loans divided by the total value of the loan portfolio (including non performing loans before the deduction of specific loan-loss provisions).

The fourteen local market indices that are used to compute returns for international index includes ATX (Austria), BAS (Belgium), OMXC (Denmark), OMXH 25 (Finland), CAC

(France), DAX 30 (Germany), ISEQ (Ireland), MIB (Italy), LuxX (Luxembourg), AEX (Netherlands), IBEX 35 (Spain), OMXS 30 (Sweden), SMI (Switzerland) and FTSE all share (United Kingdom).