Integration of market risk with credit risk measurement scenario

Integration of market risk with credit risk measurement: scenario generation through Dynamic Factor analysis A. Cipollini1 and G. Missaglia First draft: November 2004 Abstract In this paper we use stochastic simulation to measure risk associated with a bank loan portfolio. In particular, we integrate market risk with credit risk by accounting for the influence of a number of stochastic macroeconomic variables on credit events. The state of macroeconomic is measured by a Dynamic Factor. The macro-scenarios are obtained by employing Montecarlo stochastic simulation. We depart from the standard one factor model representation of portfolio credit risk. In particular, we do not consider an heterogeneous portfolio, but we also account for the impact of stochastic dependent recoveries and for the impact of two different shocks (identified as aggregate demand and supply) on portfolio credit risk. Keywords: Risk management, default correlation, Dynamic Factor JEL codes: C32,E17,G20 1 Corresponding author: Queen Mary, University of London, Mile End Road, London E1 4NS. 1 1. Introduction The proposed new Bank of International Settlement accord (known as Basel 2) provides for greater sensitivity of capital requirements to the credit risk inherent in bank loan portfolios. In light of the Basel 2 accord to reform the regulation of bank capital, there has been an extensive research on credit risk. The latter can be considered as a dominant component of risk for banks. The risk of an individual bank can be measured as the dispersion of future returns on its own portfolio. In particular, the focus of risk measurement is on downside outcomes. Consequently, measures of risk tend to focus on the likelihood of losses, rather than characterising the whole distribution of possible future outcomes. For this purpose, the analysis of credit risk has been based on the measurement of the Value at Risk ( aR), which is the potential loss that a portfolio of credit V exposures could suffer, with a predetermined confidence level, within a specified time horizon (usually, one year for a bank). A crucial input of a portfolio credit risk model, PCR, is the appropriate characterisation of default correlations to obtain the bank loan portfolio distribution with the relevant percentile (e.g. the value at risk capital requirement). Recent research suggests that defaults, and more generally the probability over upgrading, downgrading the credit quality of a borrower, vary with the business cycle. These are, for instance, the empirical findings, based upon transition matrices calculated using external ratings from Moody’s and Standard and Poor’s, of Nickell et al. (2000). Similar findings are in Bangia et al. (2000) who concentrate on the ratings of corporate borrowers and in Haldane et al. (2001) who focus on sovereign borrowers. Furthermore, the study of Jordan et al. (2002) and of Cateraineu-Rabell (2002), use transition matrices computed according to either Moody’s data or to KMW style ratings. Their findings suggest swings, across the business cycle, in the minimum capital requirements (for a portfolio of 339 loans in a shared national credit program in the United States, the former study, and for a selection of banks in G10 countries, the latter study). Other studies, based upon time series data on internal ratings suggests similar conclusions. In particular, the study of Carling et al. (2001), find a substantial fall substantial improvement in the internal ratings ver the 1994-2000 period, and consequently, a fall in the capital charge of a large Swedish bank. This was found to be associated with the gradual improvement of the Swedish Phone: +44 20 78825546. E-mail: a.cipollini@qmul.ac.uk 2 economy after the financial problems of the early 1990s. Furthermore, Segoviano and Lowe (2002), having access to time series data on the ratings assigned by a number of Mexican banks to business borrowers, find large swings in required capital. The impact of ratings migration (associated with the business cycle) on capital requirements has also received some attention recently. On the one hand, Carpenter et al. (2001) conclude that in the Untied States there is very little cyclical impact on capital charges; on the other hand, Ervin and Wilde (2001) find large swings in the minimum capital requirements. Also, the role of uncertain recoveries is important for the determination of Credit Risk VaR. The empirical study of Hu and Perraudin (2002) shows a negative correlation between probability of default and recovery rate. This finding can, for instance, be explained by observing that both default and recovery are found to depend on the state of the macro-economy (see the work by Gupton et al., 2000 and by Frye, 2000b). 2 In line with the aforementioned empirical findings, portfolio credit risk models account for the influence of the state of the business cycle on credit risk. A number of studies based on the assumption of homogeneous portfolio, and by allowing dependence of both defaults and recovery on a common factor, have provided an analityc solution for the limiting portfolio distribution (see below). However, given the heterogenous nature of the portfolio under examination in this paper and the stochastic nature of recoveries (dependent on defaults), we use stochastic simulation in order to obtain the portfolio loss distribution. For this purpose, we follow the method put forward by Krenin et al. (1998) by generating scenarios through stochastic simulation to determine conditional default probability and conditional portfolio loss distribuion. It is important to observe that we concentrate only on a “default mode” model, that is the model measures credit lossess arising exclusively from the event of default. The novel aspect of the paper are described as follows. First, the generation of scenarios occurs through a Dynamic Factor model specification. Secondly, we also depart the standard one factor model representation of portfolio credit risk. In particular, we do not consider an heterogeneous portfolio, but we also account for the impact of stochastic dependent recoveries and for the impact of two of common aggregate shocks (identified as aggregate demand and supply) on portfolio credit risk. 2 It is also important to observe that the studies of Altman and Brady (2002), Altman, et al. (2003), find that not only the state of the business cycle, but also contract–specific factors, such as, seniority and collateral, seem to affect recovery rates. 3 The outline of the paper is as follows. Section 2 highlights the analytic solution method used to retrieve Credit Portfolio Loss distribution. Section 3 describes the stochastic simulation exercise using a Dynamic Factor model; Section 4 describes the empirical results; Section 5 concludes. 2. Credit Portfolio Loss Distribution The credit portfolio loss L is given by: L = ∑ ( Dj * L j ) j =1 N (1) where N is the number of counterparts, Dj is a default indicator for obligor j (e.g. it takes value 1 if firm j defaults, 0 otherwise). Furthermore, the loss from counterpart j is given by: L j = ∑ EADhj * LGDh j h =1 H (2) where EADhj is the exposure at default to the h business unit of obligor j. Finally, LGDhj is the corresponding loss given default (equal to one minus the recovery rate, see below). Since L is a random variable, it is crucial to retrieve its probability distribution to measure portfolio credit risk. For this purpose, from (1) and (2) we can observe that we need to consider as a random variable, at least one from Dj, EADhj, and LGDhj. In this paper, in line with most of the credit portfolio risk studies, we concentrate only on the stochastic nature of defaults and loss given defaults, treating exposures as deterministic. In order to integrate credit risk with market risk, we let a credit event depend on a specific scenario x, (for instance, the state of the business cycle). Consequently, equations (1) and (2) can be replaced by: L( x) = ∑ ( D j ( x) * L j ( x )) j =1 N (1’) L j ( x ) = ∑ EADhj * LGDh j ( x ) h =1 H (2’) 4 Finally, in the Portfolio Credit Risk Analysis, the moments and the percentiles of the loss distribution of particular interest are the expected loss (e.g. the sample mean of the overall distribution), the difference between the 99% percentile and the expected loss, which gives the unexpected loss. If the forecast horizon is a year, then the unexpected loss predicts the minimum loss that can occur in one out one hundred years. Finally, if the minimum loss occurs, then its amount is predicted by the expected shortfall, computed as the mean of the distribution values beyond the 99% percentile. 2.1 Conditional probability of default: one factor model specification In this subsection we consider only uncertainty on defaults. The estimation of the probability of default for obligor j, PDj, conditional on the realisation x of a random variable X (parodying, for instance, the state of the economy), is obtained by considering the standard structural form credit risk model of Merton (1974). According to Merton (op. cit.), a firm defaults when its asset value index falls below a threshold. Specifically, define Aj as the level of firm j’s asset value index, which proxies the creditworthiness of obligor j. Let Dj symbolise the default event of firm j, then we can observe that: if Aj < cj, then Dj = 1; Dj = 0 otherwise. Given the long-term (unconditional) probability of default of firm j, PDj = P(Aj < cj) = F (cj), where F is the cumulative standard normal probability distribution, then the level of threshold cj is given by F -1(PDj). In order to estimate the (conditional) probability of default for firm j, we need to specify a stochastic process for Aj. It is customary to consider the following specification for the dynamics of the level of firm j’s asset value index: Aj = pX + 1 − p X j (3) where X is a systematic risk factor affecting simultaneously every firm and Xj is an idiosyncratic (firm specific) risk factor. Equation (3) is the classical one factor model specification for credit portfolio risk (see Schonbucher, 1998 for a detailed description). Although the different portfolio credit risk developed so far appear quite different on the surface, recent theoretical work has shown an underlying mathematical equivalence (in terms of the specification given by equation 2) among 5 them, (see Gordy, 2000). The two risk factors are assumed to have independent standard normal distributions, implying that Aj has a standard normal distribution. The parameter p measures the effects of the systematic factor on the whole set of different obligors. The conditional probability of default for firm j, e.g. the probability of the asset value falling below the threshold a realisation x of the common factor X is given by: P{[ Ajt < Φ −1 ( PD j )]| X = x} =   Φ −1 ( PD j ) − px  Φ −1 ( PD j ) − px  = P[ px + 1 − p 2 X j < Φ −1 ( PDj )] = P  X j <  = Φ      1− p2 1 − p2     (4) The conditional probability of default for the individual obligor j is a crucial input in the retrieving the joint default probability, and, consequently, the portfolio loss. In particular, under the assumption of homogenous portfolio, and deterministic loss given default, analytic solution for the asymptotic cumulative distribution and density of the portfolio loss are provided in Schonbucher (1998) 3 . Finally, Wehrspohn (2003) provides analytic closed form solution of the limiting distribution and density of the credit portfolio loss, under the classical one factor model representation and the assumption of a moderately heterogeneous portfolio and deterministic recovery rate. 2.3 Stochastic defaults: multifactor dependence In this paper, in line with the Asset Pricing Theory model developed by Ross (1976), the creditworthiness index (proxied by the corresponding stock return) which describes the financial health of obligor j is related to a set of macroeconomic factors as follows: Aj = ∑ βk X k + σ j X j k =1 r (5) 3 The Basel II proposal (as of January 2001) for the determination of minimum capital equirement is based upon the classical one factor model specification for an homogenous portfolio. In particular, a (worst case) realisation x of the common factor is given by -2.57, e.g. the 99th percentile of the Normal distribution, and it is assumed that for each borrower a proportion of 20% of the total variance is explained by the systematic factors, e.g. p = 0.2. 6 The first addend of (5) describes the systematic component of creditworthiness index Aj, entirely explained by a set of macro-variables (e.g. the credit drivers) Ak orthogonal to each other. The second addend in (5) captures the idiosyncratic component of the creditworthiness index, X j (iid and orthogonal to the macro factors Zk) with σ j = 1 − ∑ β k . Given that both the common and the k =1 r idiosyncratic factors are standard normal variables, the creditworthiness index of firm j is standard gaussian itself. Conditional on a scenario x, the probability of default according to the Merton model is now: r  α j − ∑ βk Xk  r k =1 Pr( Aj < α j | X ) = Pr( ∑ β k X k + σ j X j < α j ) = Pr  X j < σj  k =1   r   α j − ∑ βk Xk   k =1  = Φ σj             ` (6) 2.3 Stochastic recovery Recently, few studies, under the homogeneous portfolio assumption, derive an analytic solution for the credit portfolio distribution, allowing for stochastic recoveries. In particular, the dependence between the default events and losses given default is introduced through a single factor that drives both default events and recovery rates. The recovery rate is then modelled by specifying the collateral value distribution. For instance, Frye (2000a) uses a Gaussian collateral value, whereas Pykhtin (2003) focuses on a log normal distribution. However, to our knowledge, at the industry level, the computation of Credit Risk Portfolio VaR is obtained treating the recovery rate either as deterministic or as stochastic, but independent from the probability of default. The only exception is the study of Altman et al. (2002). The authors (op. cit.), using a stochastic simulation method, draw the innovations underlying recoveries from the beta distribution. In the present paper we follow the suggestion of Altman et al. (op. cit.) using a beta distribution to model stochastic recoveries. Furthermore, we impose a perfect rank correlation between the LGD and the default rate associated with the common shock scenarios. For example, when the common shock scenarios produce the largest number of defaults, the recovery rate takes 7 the smallest value. On the other hand, when the common shock scenarios produce the smallest number of defaults, the recovery rate takes the largest value. 3. Stochastic simulation In this paper, given that the heterogeneous portfolio under investigation and given recoveries stochastic and dependent on default events, we use Montecarlo simulation for the purpose of Credit Risk VaR analysis. In particular, in line with the methodology proposed by Krenin et al. (1998), in this paper, the probability of a credit event (specifically, a default probability) is obtained by conditioning on a scenario. We define a scenario as set of states of the world (e.g. a complete specification at a point in time of the relevant credit drivers) over time 4 . Conditional on a scenario, although the different obligors creditworthiness indices are correlated with each other (though their systemic component), all probabilities of defaults and (of rating changes) are independent. In order to generate a scenario, we need a model that specifies the stochastic feature of the macro economic factors. In the Credit Portfolio View approach developed by Wilson (1997), the credit drivers are modelled independently from each other and they are assumed to follow an AR(2) process. More recently, in order to account for credit drivers interdependencies, Pesaran et al. (2003) have suggested a VAR model specification for a large dataset of macroeconomic variables. In this paper, to avoid the small sample problems when using a VAR model for large dataset, the scenarios are generated using a Dynamic Factor model specification for the credit drivers (see below). It is important to observe that in the portfolio under examination (see below), some obligors sharing common features are aggregated in clusters. Each of these clusters contains a large number of obligors, each with a small contribution. In order to estimate the conditional losses for each cluster, we apply the Law of Large Numbers, hence the whole distribution collapses into a single value: the corresponding expected loss. As for the large number of obligors organised in non-clusters and given their relative large exposure, we use Montecarlo simulation to obtain the corresponding conditional portfolio loss distribution. Finally, in each scenario, the sum of losses deriving from default of the non cluster obligors (obtained through simulation) and the expected loss from the clusters gives the (conditional) portfolio loss distribution. Finally, the Montecarlo simulation has 8 been based upon the simplifying assumptions that: a) we do not account for the use of financial collateral and of credit risk mitigation techniques; b) we consider the year as the reference temporal horizon; c) we do not consider claims maturing in a period less than a year. 3.1. Scenario generation through a Dynamic Factor model Bucay and Rose (2000) have suggested the generation of scenarios through Principal Component Analysis for credit risk modelling. However, in their analysis, the extraction of factor relies on a static modelling approach, which, we argue, does not capture adequately the intrinsic dynamic structure of the factors underling the large number of credit drivers that we consider. 3.1.1 Retrieving the reduced form static factors Using equation (5), once we estimate (by OLS) the sensitivities ß, we are able to project the creditworthiness indices. This is possible once we obtain the (static) principal components from a Dynamic Factor model for a large number of credit drivers (see Stock and Watson, 2002). More specifically, a Dynamic Factor model can described along the lines of Forni et al. (2003), where the dynamics of a panel of credit drivers x t is explained by q common shocks, ut : xnt = B n( L)u t Where for each cross section unit at time t: xit = ∑ bih ( L )uht + ξit h =1 q (7) where bi(L) is a polynomial in the lag operator of order s. Following Stock and Watson (2002), we can write (7) in a static form as: xnt = An ft + ξ nt (8) 4 In a single period model there is a direct correspondence between a state of the world and a scenario; in a multi period model a scenario corresponds to a path of states of the world over time. In this paper, the forecast horizon is a year, hence, we consider a single period model. 9 where f t = ( ut' An = (a1' ut' −1 ... ut'− s ' are the static factors (which are equal to r=q(s+1)) and ) ' ' n a2 ... an ) ' = B0 ( B1n ... Bsn . The corresponding reduced form static factors are ) given by gt = Gf t, where G is a non singular matrix. The static factor space can be consistently estimated by a two stage method, which can either be the generalised principal component estimator proposed by Forni et al. (2000) or the principal component estimator proposed by Stock and Watson (2002) 5 . In this paper we use the procedure proposed by Stock and Watson which gives a consistent estimator of gt in terms of the first r principal componenents of the panel x nt : g t = Wn' xnt ^ (9) where Wn is the n×r matrix having on the columns the eigenvectors corresponding to the first r largest eigenvalues of the covariance matrix of x nt. 4.2 Retrieving the structural form common shocks From (7) we can observe that: f t = Fft −1 + et  ( q0sq) × where: F =   I  ( sq×sq) (10) ( q ×q) 0   u   , and et =  t  is orthogonal to f t-1. It follows that gt has the following 0  ( sq×1)  0    ( sq ×q )  VAR(1) representation: g t = GFG −1g t−1 = Dgt −1 + ε t (11) and the residual et is given by: ε t = Get = Gqut = (G q H ') Hut = KMHu t (12) where: 5 More recently, Kapetanios and Marcellino (2003) have proposed an alternative method, based on a state space model 10 1) Gq is the r×q matrix formed by the first q columns of G; 2) M is the diagonal matrix having on the diagonal the square roots of the q largest eigenvalues of covariance matrix of the residuals et. 3) K is the r×q matrix whose columns are the eigenvectors corresponding to the q largest eigenvalues of covariance matrix of the residuals et. 4) H is a q×q rotation matrix. Inverting the VAR we get the impulse response profile: g t = ( I − DL) −1 KMHu t (13) The identification of the q structural form common shocks is obtained by retrieving the rotation matrix H. In case of two common shocks, H is given by: cos θ H=  sin θ − sin θ  cos θ   The proposed identification scheme for the dynamic factor model used in this paper is along the lines of Giannone et al. (2002) and of Forni, et al. (2003). The authors (op. cit.) use long-run restrictions as suggested by Blanchard-Quah (1989). In particular, we consider as the first credit driver, a real variable: the industrial production, and all the remaining ones are nominal-financial variables. In line with Blanchard-Quah (op. cit.), if we assume that, contrary to the second shock, the first one does not have any long-run impact on the real industrial production, then we are able to identify the two shocks, as aggregate demand and supply common shocks, respectively. Specifically, the (exact) identification is obtained by finding the angle ? for which a zero exclusion restriction on the element corresponding to the first row and first column of the long-run multiplier matrix Cn (I-D)KMH holds. 3.1.2 Forecasting the credit worthiness index From the VAR(1) in (9) we derive the one step ahead projection: g t+ h = ε t + h + Ψ1ε t + h− 1 + Ψ 2ε t + h− 2 ... + Ψ h−1ε t +1 + Ψ h gt to estimate a large dimensional Dynamic Factor model. 11 (13) where from (12) Ψ h = [ D ( KMH )]h and et+h = KMHu t. Given that the credit drivers used in this paper are observed at monthly frequency and the forecast horizon, h = 12 and we need to project the credit drivers 12 step ahead. Specifically, x n ,t+12 is given by Cn gt+12 (where an estimate of Cn is obtained using the OLS estimator ( g t' g t ) ( gt' xnt ) ). Finally, −1 recalling (5), we can obtain the i-th forecasted credit-worthiness index by computing ßiCn gt+12, where is associated r-dimensional vector of sensitivities. 4. Empirical analysis 4.1 Data We consider a corporate portfolio, describing the exposures of an Italian bank towards small and medium sized enterprise, SME. Specifically, in this portfolio, there are 270.000 claims which according to the different type of instruments (such as receivables, trade credit loans, and financial letters of credit) are associated with 150.000 counterparts , which gives 53 billions Euro regarding the committed amount and 31 billions Euro regarding the drawn amount. The obligors with marginal exposure have been grouped in homogenous clusters in terms of rating and economic sector. This allows to consider a portfolio with 9912 obligors ( with cluster and non-clusters) which gives a total exposure of 44 billions of Euro . To summarise, we consider an heterogeneous portfolio consisting of 9912 subportfolios, with obligors treated identically within each subportfolio, but with probability of default, exposure and sensitivity differing across the subportfolios. The data span (monthly frequency) under investigation corresponds to the period after the introduction of EMU, starting in January 1999 and ending in May 2003.We now describe the data regarding the proxy for the creditworthiness index and for the credit drivers. Given that most of the obligors are non floated in the stock market, we assemble the counterparts in twenty large clusters corresponding to the following Italian MIB sub-sectors stock price indices: Food/Grocery, Insurance, Banking, Paper Print, Building, Chemicals, Transport/Tourism, Distribution, Electrical, Real Estate, Auto, Metal/Mining, Textiles, Industrial Miscellaneous, 12 Plants/Machinery, Financial Services, Finance/Part, Financial Miscellaneous, Public Utility, Media. The returns on these stock indices are used to proxy the creditworthiness indices of each obligor. The credit drivers considered are given by the real (seasonally adjusted) industrial production index and a number of nominal-financial variables. The nominal variables are CPI price index for all item, and for different following aggregate goods: clothing and footwear, communications, education, electricity and other fuels, energy, food, furnishing, health, restaurants and hotels, insurance, recreation, transport. We also include the Producer Price Index, PPI. More specifically, we consider the PPI total excluding construction, and for the following man aggregates: capital goods, durables, energy, intermediate goods, manufacturing, non- durables. The financial variables are, first, given by the term structure of interest rates in Italy and in the US. In particular, as for the short-term maturity, we consider the one, two, three, six, nine, twelve months Italian interbank rates, and the one to twelve month interbank rates in the US. As for long maturities, we consider the MSCI Italian and US government bond yields for the following maturities: one to three years; three to five years; five to seven years; seven to ten years; over ten years. Furthermore, we consider the returns on MSCI stock price indices for a number of individual countries6 and for a number of sectors (e.g. Energy, Materials, Industrials, Consumer Discretionary, Consumer Staples, Health, Financials, Information Technology, Telecommunications, Utilities) corresponding to different geographical areas (World, US, Europe, Emerging Markets). 4.2 Credit risk measurement According to the method suggested by Bai-Ng (2002), the number of static factors r is found to be equal to five 7 . As explaned above, we consider only one structural form common shock which can be interpreted as a market risk shock. As for the estimation of the sensitivies, from the Asset Pricing Theory (APT) regressions estimated by OLS (see Ross, 1976), the corresponding R2 (see Table 1) indicate that a variation in the credit drivers expain between 10% and 70% of the total variation of the Italian sub sector stock returns 8 . With the estimated principal components serving as the 6 As for developed countries we consider the SP500 and Nasdaq for the US, and the indices for France, Germany, Italy, UK, Japan, Switzerland. As for the emerging countries, we consider the indices for: Argentina, Brazil, Chile, China, Colombia, Czech Republic, Egypt, Hungary, India, Indonesia, Israel, Jorndania, Korea, Malaysia, Mexico, Morocco, Pakistan, Peru, Philippines, Poland, Russia, South Africa, Taiwan, Thailand, Turkey, Venezuela. 7 In particular, we consider the log version of the Bai-Ng method. 8 It is important to observe that the systemic component of the return series should be N(0,1) (see ). Consequently, define β and R2 the matrix of coefficients measuring the sensitivies in each APT regression and the coefficient of determination of each APT rgression, respectively. Then, for the purpose of standardising, we consider β′g t+h /(βgt+hg’t+h β’) and we multiply this ratio by R2 . Whrereas we attach weight 1- R2 to the idiosincratic component of the return series. 13 economic scenario generator and the fitted APT regressions, we simulated the loss distributions for the twelve month horizon (see description above). For this purpose, we carried out 1000 simulations for each scenario, and conditional on each scenario we carried out 1000 simulations for the idiosyncratic component of each obligor creditworthiness index3 . This gives one milion of portfolio loss realisations and by ordering we are able to obtain the unconditional loss distribution. As we can observe (see Fig. 1-6) the shape of the unconditional loss distrubution is asymmetric and highly skewed to the right. From the Figures below and Tables 2 and 3 (are values are in milions of Euro) we can draw the following conclusion. First, consider the case of a single common shock with constant recovery9 . If we compare the expected loss and unexpected loss values obtained though parametric computation (according to IRB method suggested by Basel 2) which amounts to 330 millions of Euro and 2.600 millions of Euro, then we obtain similar results by using stochastic simulation (see second column of Table 1). Secondly, by comparing results in Tables 2 and 3, we can observe that, as expected, the consideration of stochastic dependent recovery shifts to the right the unconditional loss distribution, implying high values for the expected loss, unexpected loss and expected shortfall. Finally, If we disentangle the common shock in two structural shocks: aggregate demand and aggregate supply, then we can observe that the supply shock has an higher impact on the expected loss, unexpected loss and expected shortfall. This holds for both the case of constant and of stochastic dependent recovery. 5. Conclusions Since default probabilities are driven primarily by how different obligors are tied to business cycles and to the degree of macro financial imbalances in the economy, in this paper we attempt to integrate market risk with credit risk. For the credit portfolio component of our model we use a Merton type framework, whereas the different market risk scenarios are described by few principal components obtained by fitting a Dynamic Factor model to a large number of credit drivers. For the purpose of credit risk measurement, we obtain conditional and unconditional credit portfolio 4 The asymmetric unimodal shape of the distribution in presence of an heterogenous portfolio can be explained in terms of relative low values for the (standardised) sensitivities from the analytical findings of Wehrspon (2003). Specifically, the author (op. cit.) show that in presence of a portfolio heterogenous in terms of (unconditional) probabilities to default, exposures and sensitivities, a multimodal shape of the uconditional distribution arises only in presence of an high sensitivities values. 9 In this case, we use for each obligor the same recovery rate equal to 55% equal to the one suggested by Basel for claims without guarantee; 14 distribution using Montecarlo simulation. The empirical results suggests, in general, that, in line with Altman et al. (2002), that ignoring the main feature of recoveries, as stochastic and dependent on default, can imply under serious under provision of minimum capital requirements. Furthermore, once we depart from the standard one-factor model representation of portfolio credit References E. I. Altman and B. Brady (2002): “Explaining aggregate recovery rates on corporate bond defaults”. New York University Salomon Center working Paper. E.I Altman, B. Brady, A. 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(1997a): “Portfolio Credit Risk, Part I”, Risk 10 (9), 111-117 Table 1: R2 from the APT regressions Food/Grocery Insurance Banking Paper Print Building Chemicals Transport/Tourism Distribution Electrical Real Estate Auto Metal/Mining Textiles Industrial Miscellaneous Plants/Machinery Financial Services Finance/Part Financial Miscellaneous Public Utility Media 0.17 0.45 0.61 0.24 0.55 0.42 0.31 0.34 0.64 0.41 0.37 0.26 0.51 0.09 0.42 0.42 0.51 0.38 0.66 0.71 Table 2: credit risk measures with constant recovery Expected Loss Unexpected Loss Expected Shortfall Common shock 356.02 2632.73 3831.77 demand shock 338.87 2800.75 4240.08 supply shock 371.44 2982.52 4430.75 Table 3: credit risk measures with stochastic dependent recovery Expected Loss Unexpected Loss Expected Shortfall common shock 571.72 3093.90 4641.13 demand shock 532.16 3321.54 5132.77 supply shock 591.88 3516.66 5367.63 17 Figure 1: Unconditional loss distribution: common shock and constant recovery 700000 600000 500000 400000 300000 200000 100000 0 0 1000 2000 3000 4000 5000 6000 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera Probability 356.0206 179.9219 5872.163 26.00034 527.8089 4.260533 26.68127 26392120 0.000000 Series: ULOSSC Sample 1 1000000 Observations 1000000 Figure 2: Unconditional loss distribution: demand shock and constant recovery 800000 700000 600000 500000 400000 300000 200000 100000 0 0 1000 2000 3000 4000 5000 6000 7000 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera Probability 338.8726 139.6597 7698.124 34.92575 586.7212 4.876577 35.44274 47818987 0.000000 Series: ULOSSC_DEMAND Sample 1 1000000 Observations 1000000 Figure 3: Unconditional loss distribution: supply shock and constant recovery 700000 600000 500000 400000 300000 200000 100000 0 0 1000 2000 3000 4000 5000 6000 7000 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera Probability 371.4462 180.2202 7677.598 25.35907 604.3437 4.943896 35.97148 49370276 0.000000 Series: ULOSSC_SUPPLY Sample 1 1000000 Observations 1000000 18 Figure 4: Unconditional loss distribution: common shock and stochastic dependent recovery 350000 300000 250000 200000 150000 100000 50000 0 0 1000 2000 3000 4000 5000 6000 7000 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera Probability 571.7250 370.0633 7040.595 57.60154 644.8009 3.779377 22.66184 18488447 0.000000 Series: ULOSSSTD Sample 1 1000000 Observations 1000000 Figure 5: Unconditional loss distribution: demand shock and stochastic dependent recovery 500000 Series: ULOSSSTD_DEMAND Sample 1 1000000 Observations 1000000 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera Probability 0 1250 2500 3750 5000 6250 7500 8750 532.1658 288.8718 9229.883 77.37915 715.1782 4.488810 31.33911 36820949 0.000000 400000 300000 200000 100000 0 Figure 6: Unconditional loss distribution: supply shock and stochastic dependent recovery 400000 Series: ULOSSSTD_SUPPLY Sample 1 1000000 Observations 1000000 300000 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera Probability 0 0 1250 2500 3750 5000 6250 7500 8750 591.8828 370.5046 9205.273 56.18385 733.5803 4.539785 31.92125 38286559 0.000000 200000 100000 19 20