Dependence of default probability and recovery rate in structural credit risk models: Case of Greek banks

Transcription

1 Dependence of default probability and recovery rate in structural credit risk odels: Case of Greek banks Abdelkader Derbali, Laia Jael To cite this version: Abdelkader Derbali, Laia Jael. Dependence of default probability and recovery rate in structural credit risk odels: Case of Greek banks. Journal of the Knowledge Econoy, Springer, In press, <0.007/s >. <hal > HAL Id: hal Subitted on 30 Jan 208 HAL is a ulti-disciplinary open access archive for the deposit and disseination of scientific research docuents, whether they are published or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de docuents scientifiques de niveau recherche, publiés ou non, éanant des établisseents d enseigneent et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Dependence of default probability and recovery rate in structural credit risk odels: Case of Greek banks Abdelkader Derbali* Higher Institute of Manageent of Sousse, Departent of Finance, University of Sousse, Tunisia 22 street Zarkaa Al Yaaa, Erriadh City, Sousse 4023 Eail: *Corresponding author Laia Jael Faculty of Econoic Sciences and Manageent of Sousse, Departent of Econoics, Sousse University, Tunisia Eail: Abstract: The ain idea of this paper is to exaine the dependence between the probability of default (PD) and the recovery rate (RR). For the epirically ethodology, we use the bootstrapped quantile regression and the siultaneous quantile regression for a saple of 7 Greece banks listed in Athens Exchange over the period of study fro January 02, 2006 to Deceber 3, 202. The easureent of this dependence is deterinate by using 7 indicators such as; the probability of default, the recovery rate, the nuber of defaults, the expected value of losses, the growth rate of GDP in Greece and three duy variables (the exit of another fir of the Athens Exchange, the new fir is listed in the Athens exchange and the date of the failure of Greece). The ain epirical results show that the probability of default and the recovery rate are inversely related. Based on this result, the banks are obliged to axiize their recovery rate to reduce their probability of default. Key words: probability of default; recovery rate; nuber of default; expected value of losses; bootstrapped quantile regression; siultaneous quantile regression JEL Classification: C4; C5; G2; G2; G32. Biographical notes: Dr. Abdelkader Derbali is an Assistant Professor in Finance at the Higher Institute of Manageent of Sousse in University of Sousse, Tunisia. He is one of the Editorial Board Mebers in African Journal of Accounting, Auditing and Finance, African Journal of Econoic and Sustainable Developent the Journal of Energy Markets, International Business Review, Energy Econoics, Cogent Econoic & Finance, and

3 International Review of Applied Econoics. His research interests include Risk Manageent, Systeic risk, International finance, Capital arkets and institutions, Banking and arket icrostructure, Islaic Finance, energy econoics, and financial econoics. He has published articles, aong others, in Research in International Business and Manageent, Cogent Business & Manageent, Cogent Econoic & Finance, Journal of Chinese Governance, The Chinese Econoy, Journal of Energy Markets, African Journal of Accounting, Auditing and Finance, International Journal of Econoics and Accounting, International Journal of Critical Accounting, and International Journal of Trade and Global Markets. Dr. Laia Jael is a PhD in Econoics at the Faculty of Econoic Sciences and Manageent of Sousse, Tunisia. His research interests include Econoic analysis, financial econoics, quantitative finance, financial developent, and energy coodities. He is one of the Editorial Board Mebers in the Cogent Econoics and Finance, Journal of Chinese Governance, International Journal of Risk and Contingency Manageent, International Journal of Sustainable Econoies Manageent, International Journal of Food and Beverage Manufacturing and Business Models, and International Journal of Managerial Studies and Research. She has published articles in African Journal of Accounting, Auditing and Finance and in Cogent Econoic & Finance. 2

4 . Introduction This literature review briefly recapitulates the way credit risk odels, which have studied during the last thirty years, treat Recovery Rate and, ore specifically, their relationship with the Probability of Default of fir. These odels can be divided into two ain categories (Atan et al., 2002) such as, Credit pricing odels and Portfolio credit value-at-risk (VaR) odels. Thus, credit pricing odels can in turn be divided into three ain approaches as, First generation structural-for odels, Second generation structural-for odels, and Reduced-for odels. These three different approaches, together with their basic assuptions advantages, drawbacks and epirical perforance as, First generation structural-for odels: the Merton approach, Second generation structural-for odels, Reduced-for odels, Credit value-atrisk odels, and soe latest contributions on the PD-RR relationship. It has been noted that default probabilities and default rates and average recovery rates are negatively correlated (Altan et al., 2005). Then, both variables also see to be driven by the sae coon factor that is persistent over tie and clearly related to be the business cycle: in recessions or industry downturns, default rates are high and recovery rates are low. Thus, the ain idea for this study is to answer the question follows: As the Probability of Default is depended to the Recovery Rate and conversely. Then, we use the bootstrapped quantile regression and the siultaneous quantile regression for a saple coposed of 7 Greece banks listed in Athens Exchange during the period through January 02, 2006 to Deceber 3, 202. To estiate this dependence, we utilize 7 indicators such as; the probability of default, the recovery rate, the nuber of defaults, the expected value of losses, the growth rate of GDP in Greece and three duy variables (the exit of another fir of the Athens Exchange, the new fir is listed in the Athens exchange and the date of the failure of Greece). The ain epirical findings deonstrate that the probability of default and the recovery rate are inversely correlated. Based on these findings, the banks are obliged to axiize their recovery rate to decrease their probability of default. Then, the rest of this paper is structured as follows: The literature review is developed on section 2. The dependence between the probability of default and the recovery rate is presented in section 3. In section 4, we describe the data, the econoetric ethodologies and PD : the Probability of default, RR: The Recovery Rate. 3

5 the odel used in this paper. Section 5 discusses and analysis of the econoetric findings. Finally, the sixth section is considered to conclude. 2. Literature review The ain idea of this study is to describe and to deterinate the relationship of dependence between probability of default and the recovery rate. In the literature, this dependence is treated by any authors. The table shows the ain works that developed and studied the dependence between the PD and the RR. Table : The treatent of the dependence between PD and RR First generation structural-for odels Second generation structural-for odels Main odels and Treatent of PD related epirical results Credit Pricing odels Merton (974), Black and Cox (976), Geske (977), Vasicek (984), Crouhy and Galai (994), Jones et al. (984). Ki et al. (993), Nielsen et al. (993), Hull and White (995), Longstaff and Schwartz (995). Reduced-for odels Litteran and Iben (99), Madan and Unal (995), Jarrow and Turnbull (995), Jarrow et al. (997), Lando (998), Duffie and Singleton (999), Duffie (998) and Duffee (999). Latest contributions on the PD-RR relationship Frye (2000), Jarrow (200), Carey and Gordy (200), Altan and Brady (2002). PD and RR are a function of the structural characteristics of the fir. RR is therefore an endogenous variable. RR is exogenous and independent fro the fir s asset value. Reduced-for odels assue an exogenous RR that is either a constant or a stochastic variable independent fro PD. Both PD and RR are stochastic variables which depend on a coon systeatic risk factor (the state of the econoy). Credit value at risk odels Relationship between PD and RR PD and RR are inversely related. RR is generally defined as a fixed ratio of the outstanding debt value and is therefore independent fro PD. Reduced-for odels introduce separate assuptions on the dynaic of PD and RR, which are odeled independently fro the structural features of the fir. PD and RR are negatively correlated. In the acroeconoic approach this derives fro the coon dependence on one single systeatic factor. In the icroeconoic approach it derives fro the supply and deand of defaulted securities. CreditMetrics Gupton, Finger and Bhatia Stochastic variable (beta RR independent fro PD (997) distr.) CreditPortfolioView Wilson (997a and 997b). Stochastic variable RR independent fro PD CreditRisk+ Credit Suisse Financial Constant RR independent fro PD Products (997). KMV CreditManager McQuown (997), Crosbie (999). Stochastic variable RR independent fro PD 3. Dependence between the probability of default and the recovery rate: analytical analysis 4

6 By analyzing the previous literature, we can conclude that the default probability was estiated according to several approaches. So, the default probability can be estiated by basing itself on historical series of default by easuring the risk by the rating or the score (Altan, 968). Epirically, the easures of score call on to alternatives as the analysis in ain coponent, the logistic regression and the Probit analysis. Jonkhart (979) deducts the default probability fro the spreads of rates available on arkets. The works of Jonkhart are carried out by Iben and Litteran (989), Wu and Yu (996), Altan (988, 989), Asquith et al. (989), Rosenberg and Gleit (994), Hand and Henely (997) and Thoas (2000), who deducted the default probability fro historical data on the bonds having been lacking by type of rating and by type of ter. Merton (979), Black and Scholes, Black and Cox (976), Geske (977) and Lee (98) deducted the default probability fro the volatility of assets. This ethod is intended for the highly-rated credits. In our paper, we are going to base ourselves on the analysis developed by Merton (974). So, Merton s odel is based on the hypothesis which the copany k has a certain quantity of debt with zero-coupon. The noinal value of this debt is F k and it becoes due in the aturity date T. The Fir is declared default if, in the date T, the value of its assets is lower than its noinal value, is, if V k(t) < F k. The recovery rate spells then under the following shape: R k = F k V k T () And the loss in the case of default is: L k = R k = F k V k T F k (2) By using the function of Heaviside (Θ), we can deterinate the loss individual can be expressed by the following for: L k = V k T F k Θ V k T F k (3) So, in the Merton s odel, the losses value and the recovery rate are directly deterined by the value of assets to the date of aturity. 5

7 Thus, the stochastic odeling of the arket value of a copany V k (T) allows to evaluate his credit risk. The probability density function (pdf) of arkets values in the date of aturity P (V (T)). So, the default probability is given by: Vk k F k P D,k = 0 P Vk V k T dv k T (4) And the recovery rate will be calculated as follows: R k = P D,k F k 0 V k T F k P Vk V k T dv k T Let us consider now a portfolio of credit K, where the arket value of every copany is correlated in one or in several variables. Under the condition of the realizations of variables, we obtain different values of P D,k and of R k (5). In fact, we can deonstrate a functional dependence between the probability of default and the recovery rate. This is in contrast with what is evoked by certain surrounding areas of odeling which suppose the existence of an independence of these qualities. By supposing the existence of a process of underlying distribution of the value of the copany, we can easily deduct all the results relative to the easure of the probability of default and the recovery rate. So, we consider a hoogeneous portfolio of size K. The noinal value of this portfolio is Fk F and the first arket values are Vk (0) V0. The evolution in the tie of the arket value of a single fir k is odeled by a stochastic differential equation of the following for: dv k V k = μdt + cσdw + c σdw k (6) This equation describes a process of correlated diffusion to a deterinist ter dt and a correlated linearly diffusion. The paraeters of this process are; the constant, the volatility σ and the correlation c between the fir return and the arket return. The process of Wiener indicated by dw k and fluctuations respectively. dw, describes the idiosyncratic fluctuations and the arket 6

8 So, the evaluation of the prices of the options on the financial arket is based on two paraeters iportant to know the volatility and the fluctuations of assets (Gatfaoui, 2006; Giovanni et al., 202). In this aligned, the fluctuations in the prices of assets are understandable by two different and independent risk factors which are the systeatic factor and the idiosyncratic factor. Therefore, the pricing of assets is iproved there because the distortions of the price of the underlying are decoposed into two parts: A coponent of arket volatility steing fro systeatic fluctuations in the price of asset ( dw ). A coponent of idiosyncratic volatility steing fro specific fluctuations in the price of asset ( dw ). k T For increent of discreet tie t= N, where the tie is divided on N stages, we arrive at the discreet forulation of the stochastic differential equation above. The arket value of k firs in the aturity can be written in the following for: N V k T = V 0 + μδt + cση,t Δt + cσε k,t Δt (7) t= With, t, and kt, are independent rando variables and they follow a noral distribution law. We will try in what follows to deterine arket return N X and the recovery rate R X. On the arket return D average yield of all k firs over a period of tie to aturity. X, the nuber of default R X which defines the X = K K k= V k T V 0 = K K N k= t= + μδt + cση,t Δt + cσε k,t Δt (8) For K, we can express the average on k as the value of the hope kt,. For the independence of kt, for different k and t, we can write then: N X + = + μδt + cση,t Δt + cσ ε k,t Δt (9) t= With : 7

9 ε k,t = 0 (0) Then, the expression of X + will be siplified as follows: X + = N + μδt + cση,t Δt N = exp ln + μδt + cση,t Δt t= t= exp μ cσ2 2 T + σ c t N η,t t= (0) In this stage, we apply the logarith to the function above. The rando variable t, follows a standard noral distribution. Then, we obtain: ln X + = μ cσ2 2 T + σ ct N N η,t t= () 2 c T Thus, the variable ln(x ) is norally distributed with average ( T ) and variance 2 2 c T. So, by basing itself on the noral logarithic distribution, we can write the probability density function as follows: p X X = X + 2πcσ 2 T exp ln X + μt + 2 cσ2 T 2cσ 2 T 2 (2) For a single fir k we can write: N ln V k T = ln + μδt + cση V,t Δt + cσε k,t Δt 0 t= ln X + μ cσ2 2 T + σ ct N N t= η,t (3) Thus, the arket return independent and the variable X is considered a constant. Thereafter, all variables V k ( T) are V ( ) 2 ( c) 2 ln( X ) Tand a variance ( c) T. 2 k T ln is norally distributed and we have an average V 0 8

10 Since, we considered a hoogeneous portfolio; we oit the index k in follows. This allows a better rating and effective results. The probability density function for the arket value of a fir is given by the following for: P v V T = X + 2π( c)σ 2 T exp V T ln V 0 ln X ( c)σ2 T 2( c)σ 2 T 2 (4) So, the individual probability of default is given by the following function: F P D X = p v V T dv T = Φ 0 V T ln V 0 ln X ( c)σ2 T 2( c)σ 2 T 2 (5) Where, Φ denotes the cuulative standard noral distribution. The expected value of the loss of individual default L X = P D X * ( ) L VT can be calculated as follows: F = 0 F V T F V T ln P D X Φ V 0 p v V T dv T ln X ( c)σ2 ( c)σ 2 T exp ln X + ln F V 0 Φ V T ln V 0 ln X ( c)σ2 ( c)σ 2 T (6) The expected recovery rate is: R X = L X (7) In the case of a hoogeneous portfolio, the loss of a portfolio (average loss) is obtained by the following for: L X = P D X L X (8) 9

11 For clarity, we introduce the function: A X = ln F V 0 ln X + (9) Where B is the coposite paraeter which is written as follows: B = c σ 2 T (20) However, the expressions of ( ) P X and R X D are siplified as follows: P D X = Φ A X + 2 B2 B (2) And, R X = A(X )Φ A X + 2 B2 B Φ A X + 2 B2 B (22) The relationship between the probability of default and the recovery rate does not depend on B only, but it is set by AX ( ). Thus, the paraeter B can be easured by the probability of default and the recovery rate. In addition, reversing the expression of P ( X ), we can express A in ters of P D : D A = BΦ P D 2 B2 (23) To clarify the effect of the idiosyncratic fluctuations and the arket fluctuations on the volatility, Schäfer and Koivusalo (20) proposed a relationship of functional dependence for the probability of default and the recovery rate. The recovery rate is expressed by the following for: L P D = P D exp BΦ P D 2 B2 Φ Φ P D B (24) 0

12 If should be noted that this functional relationship depends on a single paraeter B. We can see that for higher values of B lead to an overall decrease in the recovery rate. Fro the equation above R P D default probabilities:, we can obtain the functional relationship of the portfolio loss and L P D = P D exp BΦ P D 2 B2 Φ Φ P D B (25) For a high value of K K, the idiosyncratic is zero and the arket return only by the realization of the ter t, k. The nuber of default k D X is defined N X easure the nuber of ties that the inequality V ( T) F is feasible. We can estiate the value of the probability of default as follows: P D X N D X K (26) The loss of the portfolio is then obtained as the average of the individual losses: L X = K K L k k= (27) We can deduce the following relationship: The recovery rate is expressed as follows: L X = P D X R X (28) R X = L X P D X K L X N D X (29) Several studies have shown that the nuber of default N ( X ) is strictly non-zero. This is justified based on a large portfolio is easured by K. If we based on the evolution of a portfolio, we can obtain different values for the arket return ( X ), the nuber of defaults N ( X ), the probability of default P ( X ) and the recovery rate RX ( ). D 4. Data and epirical odel D D

13 In this section, we identify the sources of our data. We present the data itself and describe the regression odel. Finally, we use to investigate the relation of dependence between the probability of default and the recovery rate. 3.. Data In this paper, we eploy the indicator of 7 banks quoted in the Athens Exchange of through the period fro January 02, 2006 to Deceber 3, 202. The list of banks included in this study is presented in the Table 2. The balance sheet data is collected fro Statistical Bulletin of The Athens Exchange. In this study, we use the regression analysis to identify the dependence between PD and RR. The descriptive statistics applies to find the ean, the axiu, the iniu and standard deviation, Skweenes and Kurtosis of those variables. The Pearson correlation tests applied to deal with the probles. Table 2: List of Banks Nae of Bank The study period ALPHA ΒΑΝΚ (ΚΟ) 02/0/2006 3/2/202 ASPIS BANK (ΚΟ) 02/0/ /06/200 ATTICA BANK (ΚΟ) 02/0/2006 3/2/202 BANK OF CYPRUS (CR) 02/0/2006 3/2/202 BANK OF GREECE (CR) 02/0/2006 3/2/202 ΕΓΝΑΤΙΑ BANK (ΚΟ) 02/0/ /09/2007 ΕΓΝΑΤΙΑ BANK (ΠΟ) 02/0/ /08/2007 EMPORIKI BANK (CR) 02/0/ /04/20 EUROBANK EFG (ΚΟ) 02/0/2006 3/2/202 GΕΝΙΚΙ ΒΑΝΚ (CR) 02/0/2006 3/2/202 MARFIN EGNATIA BANK (CR) 02/0/2008 3/03/20 MARFIN FINANCIAL GROUP (ΚΟ) 02/0/ /03/2007 MARFIN POPULAR BANK (ΚΟ) 02/0/2008 /04/202 NATIONAL BANK (CR) 02/0/2006 3/2/202 PIRAEUS BANK (CR) 02/0/2006 3/2/202 PROTON BANK S,A, (CR) 02/0/2006 3/2/202 TT HELLENIC POSTBANK (CR) 02/0/ /2/ Econoetric ethodology In this study, we utilize two econoetric techniques to quantify the dependence between the probability of default and recovery rate. Those techniques are the Bootstrapped Quantile Regression and the Siultaneous Quantile Regression. Then, we eploy these two techniques because the paraeter quantile regression provides an estiate of the change in a specific quantile of the response variable produced by a unit change in the predictor. 2

14 3.3. Epirical odel In our study, we use two odels who describe the dependence between the probability of default and the recovery rate. We estiate the probability of default in function of seven variables. All these variables are explained in follows (Frye, 2000; Altan, 200; Gordy, 200; Altan et al., 2002; Altan et al., 2005; Bruche and Gonzalez-Aguado, 2008; Becker, 203). The probability of default is estiated by the odel presented as follow: (Equation ) PD t = f RR t, ND t, L t, Duy t, Duy2 t, Duy3 t When, PD t denotes the probability of default at the oent t, RR t denotes the recovery rate at the oent t, ND t denotes the nuber of default at the oent t and L t denotes the expected value of losses at the oent t. Duy t indicates that a new fir is listed in the Athens Exchange at the oent t. This variable that takes the value when a new fir is listed in the Athens Exchange and takes 0 in the opposite occur. Duy2 t indicates that a new fir is going out of the Athens Exchange at the oent t. This variable that takes the value when a new fir is going out of the Athens Exchange and takes 0 in the opposite occur. Duy3 t indicates that Greece declare his failure at the oent t. This variable takes the value after the date of failure and 0 before the date of failure. The eployed data are daily and which are collected fro the publication of the Athens Exchange. 4. Results and Discussion Within the fraework of this paper, we present a descriptive statistics analysis of the various variables used in all estiations. These variables are utilized to estiate the dependence between probability of default and recovery rate. First of all, the nuber of the observations is liited to 748 observations concerning the two odels. Table 3 shows all the descriptive statistics (ean, ax, in, the standard deviation, the Skewness and the Kurtosis) relative to variables used in the different estiation of the variable PD. According to this Table, we can reark that the axiu of probability of default is equal to through the period of study. However, the axiu of recovery rate is equal to. This finding iplies that the probability of default can be absorbed by the recovery rate. 3

15 For the two statistics of skewness (asyetry) and kurtosis (leptokurtic), we can observe that the two variables used in our study are characterized by non-noral distribution. The positive sign of the skewness coefficients indicate that the variable is skewed to the right and it is far fro being syetric for all variables in except of the recovery rate and the expected value of losses. Also, the Kurtosis coefficients confir that the leptokurtic for all variables used in this study show the existence of a high peak or a fat-tailed in their volatilities. Table 3: Descriptive Statistics Variable Obs Mean Std Div Min Max Skewness Kurtosis PD RR ND L Duy Duy Duy Then, we eploy two types of estiations, such as, the Bootstrapped Quantile Regression and the Siultaneous Quantile Regression. The choice of the two ethodologies is justified by the objective of this paper. Then, the purpose of this study is to exaine the correlation aong the Probability of Default and the Recovery Rate. Those econoetric techniques allow describing the dependence between tow variables based on their volatilities. Figure shows the evolution of the probability of default and the recovery rate for each year ( ). In Figure 2, we present the volatility of the PD and the RR through the period fro January 02, 2006 to Deceber 3, 202. Figure : The volatility of the PD and the RR (by year) 4

16 Probability of Default Recovery Rate Figure 2: The volatility of the PD and the RR Probability of Default Recovery Rate 5

17 In this study, we ade a test of the correlation between the various utilized variables. Table 4 reports the epirical results relative to the correlation. So, the epirical findings present that all the coefficient of correlation of Pearson does not exceed the liit of tolerance of 0.7, so it does not cause probles during the estiation of the odel which easures the PD. Table 4: The atrix of correlation PD RR ND L Duy Duy2 Duy3 PD.0000 RR (0.0000)* ND (0.0000)* (0.0000)* L (0.0000)* (0.0000)* (0.0000)* Duy (0.832) (0.6306) (0.63) Duy (0.2564) (0.680) (0.520) Duy (0.9469) (0.0000)* (0.0000)* Value significant in a threshold of: (*) %; (**) 5% et (***) 0% (0.762) (0.26) (0.0000)* (0.9495) (0.3850) (0.4498).0000 Also, we conduct an unit root test for tie series. Thus, we eploy the Augented-Dickey- Fuller and Phillips-Perron test. According to the results shown in Table 5 and Table 6, we find that all the calculated values of the t-student or t-statistical values are inferior to the critical thresholds of %, 5% and 0%. In this case, all the variables eployed in this paper are stationary. Table 5: The unit root test of Augented-Dickey-Fuller Variables t-statistic Critical value at % Critical value at 5% DP Critical value at 0% The hypothesis rejected H0: presence of unit root. So the variable is stationary RR H0: presence of unit root. So the variable is stationary ND H0: presence of unit root. So the variable is stationary L Duy Duy Duy H0: presence of unit root. So the variable is stationary H0: presence of unit root. So the variable is stationary H0: presence of unit root. So the variable is stationary H0: presence of unit root. So the variable is stationary Table 6: The unit root test of Phillips-Perron Variables t-statistic Critical value at % Critical value at 5% Critical value at 0% The hypothesis rejected DP H0: presence of unit root. So the variable is stationary RR H0: presence of unit root. 6

18 ND L Duy Duy Duy So the variable is stationary H0: presence of unit root. So the variable is stationary H0: presence of unit root. So the variable is stationary H0: presence of unit root. So the variable is stationary H0: presence of unit root. So the variable is stationary H0: presence of unit root. So the variable is stationary To pursue our epirical analysis, we estiate the variables PD in Table 7 and the Table 8. So, we estiate the variable PD by using 8 estiations for each of both variables and by using two econoetric techniques. In the table 7, we eploy the Bootstrapped Quantile Regression. Then, we notice that all the values of the statistical Pseudo-R² are alost equal to 0.80 in all estiates. So, we can conclude that the estiated odel is characterized by a good linear adjustent. In our odel, the probability of default is estiated based on other explicative variables. The results of estiation are presented in Table 7. This table suarizes all estiations relative to the odel (PD), we show that there are four significant variables with different thresholds. The first one, it is the variable RR, is statistically significant and negative in a % threshold in four estiations (, 2, 3 and 5), in a 5% threshold in the sixth estiation and in a 0% threshold in the last estiations. In this context, the variable RR has a negative ipact on the probability of default of the Greek banks. Then, when the recovery rate increases, the banks have profit to supply an iportant exposure to failure. We can conclude that a high recovery rate allow to absorb losses incurred by banks in Greece. The variable ND is statistically significant and positive in a % threshold in all estiations. The nuber of default of banks affects their probability of default. Thus, the high nubers of default reflect that the probability of default is high. The variable L (the expected losses) is statistically significant and negative in a 5% threshold only in all estiation. This confirs the literature, because the high value of the expected losses allows the bank to iniize their probability of default in future. For the three duy variables used in our study, we conclude that only the duy variable is significant in 5% threshold in the estiation 2 and 5. This variable affects negatively the probability of default. Then, the entry of a new bank in the Athens Exchange leads to the 7

19 iniization of the probability of default of the existing banks in the financial arket of Greece. In the table 8, we utilize the Siultaneous Quantile Regression. Then, we find that all the values of the statistical Pseudo-R² are equal to 0.80 in all estiates. Then, we can reark that the estiated odel is characterized by a good linear adjustent. In the odel (PD), the probability of default is estiated based on other explicative variables. The results of estiation are presented in Table 7. After eploying the second econoetric techniques (Siultaneous Quantile Regression), we conclude that all the results have alost the sae significance thresholds then the first econoetric ethodology. Then, for the two econoetric techniques the ipact of different variables reains the sae. Epirically, we can find that the probability of default is inversely related to the recovery rate. The recovery rate is not constant; it decreases with increasing of the probability of default. This epirical finding is confired by these figures follows. On these figures we presented the volatility of probability of default and recovery rate of all Greek banks and by year. The dependence between PD and RR is justified by the correlation coefficients of Pearson who presented in Table 4. The dependence between the Probability of Default and the Recovery rate of all banks is shows in figure 3, figure 4, figure 5, figure 6, figure 7, figure 8 and figure 9. All these Figure shows the dependence between the Probability of Default and the Recovery rate of all banks by years (see, figure 3, figure 4, figure 5, figure 6, figure 7, figure 8 and figure 9) 8

20 Table 7: Estiation by Bootstrapped Quantile Regression Dependent variable: PD Period of estiation : Explicative variables Estiation Estiation 2 Estiation 3 Estiation 4 Estiation 5 Estiation 6 Estiation 7 Estiation 8 RR (-4.35)* (-4.44)* (-4.38)* (-.85)*** (-4.42)* (-.75)** (-.8)*** (-.82)*** ND (85.20)* (86.20)* (85.3)* (98.80)* (8.03)* (98.3)* (97.62)* (97.55)* L (-.79)* (-.69)* (-.70)* (-9.98)* (-.48)* (-9.90)* (-9.99)* (-9.84)* Duy (-2.8)** (-2.8)** (-.28) (-.29) Duy (0.06) (0.06) (0.3) (0.3) Duy (.45) (.35) (.42) (.4) CONSTANT (4.70)* (4.82)* (4.79)* (.70)*** (4.76)* (.59) (.66)*** (.67)*** Nuber of obs Pseudo R² Value significant in a threshold of: (*) %; (**) 5% et (***) 0%. Table 8: Estiation by Siultaneous Quantile Regression Dependent variable: PD Period of estiation : Explicative variables Estiation Estiation 2 Estiation 3 Estiation 4 Estiation 5 Estiation 6 Estiation 7 Estiation 8 RR (-4.4)* (-4.23)* (-4.20)* (-.84)*** (-4.37)* (-.57) (-.9)*** (-.83)*** ND (83.62)* (82.2)* (84.06)* (00.93)* (82.9)* (46.53)* (99.74)* (96.00)* L (-.7)* (-.56)* (-.89)* (-0.03)* (-.66)* (-0.0)* (-0.25)* (-9.98)* Duy (-2.20)** (-2.2)** (-0.80) (-.27) Duy (0.05) (0.05) (0.30) (0.3) Duy (.37) (.42) (.42) (.4) CONSTANT (4.75)* (4.6)* (4.79)* (.66)*** (4.72)* (.6) (.73)*** (.64)*** Nuber of obs Pseudo R² Value significant in a threshold of: (*) %; (**) 5% et (***) 0%. 9

21 Figure 3: The volatility of the PD and the RR in 2006 (by Banks) ENATIA BANK (KO) ENATIA BANK (II O) ALPHA BANK ASPIS BANK ATTICA BANK BANK OF CYPRUS (CR) BANK OF GREECE (CR) EMPORIKI BANK (CR) EUROBANK EFG GENIKI BANK (CR) MARFIN FINANCIAL GROUP NATIONAL BANK (CR) PIRAEUS BANK (CR) PROTON BANK S.A. (CR) Probability of Default Recovery Rate Figure 4: The volatility of the PD and the RR in 2007 (by Banks) ENATIA BANK (KO) ENATIA BANK (II O) ALPHA BANK ASPIS BANK ATTICA BANK BANK OF CYPRUS (CR) BANK OF GREECE (CR) EMPORIKI BANK (CR) EUROBANK EFG GENIKI BANK (CR) MARFIN FINANCIAL GROUP NATIONAL BANK (CR) PIRAEUS BANK (CR) PROTON BANK S.A. (CR) Probability of Default Recovery Rate 20

22 Figure 5: The volatility of the PD and the RR in 2008 (by Banks) ALPHA BANK ASPIS BANK ATTICA BANK BANK OF CYPRUS (CR) BANK OF GREECE (CR) EMPORIKI BANK (CR) EUROBANK EFG GENIKI BANK (CR) MARFIN EGNATIA BANK (CR) MARFIN POPULAR BANK NATIONAL BANK (CR) PIRAEUS BANK (CR) PROTON BANK S.A. (CR) TT HELLENIC POSTBANK (CR) Probability of Default Recovery Rate Figure 6: The volatility of the PD and the RR in 2009 (by Banks) ALPHA BANK (CR) ASPIS BANK SA (CR) ATTICA BANK S.A. (CR) BANK OF CYPRUS (CR) BANK OF GREECE (CR) EMPORIKI BANK (CR) EUROBANK EFG (CR) GENIKI BANK (CR) MARFIN EGNATIA BANK (CR) MARFIN POPULAR BANK (CR) NATIONAL BANK (CR) PIRAEUS BANK (CR) PROTON BANK S.A. (CR) TT HELLENIC POSTBANK (CR) Probability of Default Recovery Rate 2

23 Figure 7: The volatility of the PD and the RR in 200 (by Banks) 0 ALPHA BANK (CR) ASPIS BANK ATTICA BANK BANK OF CYPRUS (CR) BANK OF GREECE (CR) EMPORIKI BANK (CR) EUROBANK EFG (CR) GENIKI BANK (CR) MARFIN EGNATIA BANK (CR) MARFIN POPULAR BANK (CR) NATIONAL BANK (CR) PIRAEUS BANK (CR) T PROTON BANK S.A. (CR) TT HELLENIC POSTBANK (CR) Probability of Default Recovery Rate Figure 8: The volatility of the PD and the RR in 20 (by Banks) ALPHA BANK (CR) ATTICA BANK S.A. (CR) BANK OF CYPRUS (CR) BANK OF GREECE (CR) EMPORIKI BANK (CR) EUROBANK EFG (CR) GENIKI BANK (CR) MARFIN EGNATIA BANK (CR) MARFIN POPULAR BANK (CR) NATIONAL BANK (CR) PIRAEUS BANK (CR) PROTON BANK S.A. (CR) TT HELLENIC POSTBANK (CR) Probability of Default Recovery Rate 22

24 Figure 9: The volatility of the PD and the RR in 202 (by Banks) ALPHA BANK (CR) ATTICA BANK S.A. (CR) BANK OF CYPRUS (CR) BANK OF GREECE (CR) EUROBANK EFG (CR) GENIKI BANK (CR) MARFIN POPULAR BANK (CR) NATIONAL BANK (CR) PIRAEUS BANK (CR) TT HELLENIC POSTBANK (CR) Probability of Default Recovery Rate We, also, use a Kruskal-Wallis equality-of-populations rank test. The Kruskal-Wallis oneway analysis of variance by ranks is a non-paraetric ethod for testing whether saples originate fro the sae distribution. It is eploy for coparing ore than two saples that are independent, or not related. The paraetric equivalent of the Kruskal-Wallis test is the one-way analysis of variance (ANOVA). The results of this test can accept the hypothesis H0. ie the average of the different types of saple studied are not significantly different (Mean Rank = 6 < Index Kruskal-Wallis = 6,73). We calculate also the Value-at-Risk on four confidence level chosen (see, Table 9). Table 9: The VaR results Confidence level VaR 95% 0, % 0, % 0, % 0, In recession periods, the nuber of defaulting banks or firs in generally rises. On top of this, the average aount recovered on the bonds of defaulting banks tends to decrease. Our paper purpose an econoetric odel in which this joint tie-variation in default rates and 23

25 recovery rate distribution by a quantile regression, which given the iportance if the volatilities. 5. Conclusion The dependence between the probability of default and the recovery rate has a crucial influence on large credit portfolio losses. Thus, the probability of default and the recovery rate are often odeled independently in current credit risk odels: KMV odel, CreditMetrics odel, CreditRisk+ and Credit Portfolio View. In this paper, we utilize the bootstrapped quantile regression and the siultaneous quantile regression for a saple coposed of 7 Greek banks listed in Athens Exchange over the period fro January 02, 2006 to Deceber 3, 202. To estiate this dependence, we utilize 7 indicators such as; the probability of default, the recovery rate, the nuber of defaults, the expected value of losses, the growth rate of GDP in Greece and three duy variables (the exit of another fir of the Athens Exchange, the new fir is listed in the Athens exchange and the date of the failure of Greece).. Finally, we conclude that the probability of default and the recovery rate are inversely related. This result is confired by those table and figure presented in the fourth section. Based on the results found in this study, the banks are obliged to axiize their recovery rate to reduce their probability of default. References Ali, A. and Daly, K. (200). Macroeconoic deterinants of credit risk: Recent evidence fro a cross country study. International Review of Financial Analysis, 9, pp Altan, E.I. (989). Measuring corporate bond ortality and perforance. Journal of Finance, 44, pp Altan, E.I., Brady, P., Resti, A. and Sironi, A. (2005). The link between default and recovery rates: effects on the procyclicality of regulation capital ratios: Theory, Epirical evidence, and iplications. Journal of Business, 78(6), pp Bensoussan, A., Crouhy, M. and Galai, D. (995). Stochastic equity volatility related to the leverage effect II: Valuation of European equity options and warrants. Applied Matheatical Finance, 2, pp

26 Berry, M., Bureister, E. and McElroy, M. (998) Sorting our risks using known APT factors. Financial Analysts Journal, 44(2), pp Crouhy, M., Galai, D. and Mark, R. (2000). A coparative analysis of current credit risk odels. Journal of Banking & Finance, 24, pp Duffie, D. and Singleton, K.J. (999). Modeling the ter structures of defaultable bonds. Review of Financial Studies, 2, pp Figlewski, S., Frydan, H. and Liang, W. (202). Modeling the effect of acroeconoic factors on corporate default and credit rating transitions. International Review of Econoics and Finance, 2, pp Grundke, P. (2005). Risk Measureent with Integrated Market and Credit Portfolio Models. Journal of Risk, 7(3), pp Grundke, P. (2009). Iportance sapling for integrated arket and credit portfolio odels. European Journal of Operational Research, 94, pp Huang, S.J. and Yu, J. (200). Bayesian analysis of structural credit risk odels with icrostructure noises. Journal of Econoic Dynaics & Control, 34, pp Jarrow, R. and Turnbull, S. (995). Pricing derivatives on financial securities subject to credit risk. The Journal of Finance, 50, pp Jarrow, R.A., Lando, D. and Yu, F. (200). Default risk and diversification: theory and applications. Matheatical Finance, 5, pp Jarrow, R.A. (20).Credit arket equilibriu theory and evidence: Revisiting the structural versus reduced for credit risk odel debate. Finance Research Letters, 8, pp Lee, W.C. (20). Redefinition of the KMV odel s optial default point based on geneticalgoriths Evidence fro Taiwan. Expert Systes with Applications, 38, pp Liao, H.H., Chen, T.K. and Lu, C.W. (2009). Bank credit risk and structural credit odels: Agency and inforation asyetry perspectives. Journal of Banking & Finance, 33, pp Merton, R. (947). On the pricing of corporate debts: the risk structure of interest rates. Journal of Finance, 29, pp

27 Musto, D.K. and Souleles, N.S. (2006). A portfolio view of consuer credit. Journal of Monetary Econoics, 53, pp Tarashev, N. (200). Measuring portfolio credit risk correctly: Why paraeter uncertainly atters. Journal of Banking & Finance, 34, pp Vetendorpe, A., Ho, N.D., Vetuffel, S. and Dooren, P.V. (2008). On The Paraeterization of the CreditRisk+ Model for Estiating Credit Portfolio Risk. Insurance: Matheatics and Econoics, 42(2), pp Wilson, T.C. (997a). Portfolio credit risk (I). Risk, 0(9), pp. -7. Wilson, T.C. (997b). Portfolio credit risk (II). Risk, 0(0), pp Xiaohong, C., Xiaoding, W. and Desheng, W.D. (200). Credit risk easureent and early warning of SMEs: An epirical study of listed SMEs in China. Decision Support Systes, 4, pp Zhang, Q. and Wu, M. (20). Credit Risk Migration Based on Jarrow-Turnbull Model. Systes Engineering Procedia, 2, pp