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Forecasting Mortgage Securitization Risk under Systematic Risk and Parameter Uncertainty Daniel R¨sch a Harald Scheule b 1 o a Institute o of Banking & Finance, Leibniz University of Hannover, K¨nigsworther Platz 1, 30167 Hannover, Germany, Phone: +49-511-762-4668, Fax: +49-511-762-4670, mailto: Daniel.Roesch@ﬁnance.uni-hannover.de b Department of Finance, Faculty of Business and Economics, University of Melbourne, Victoria 3010, Australia, Phone: +61-3-8344-9078, Fax: +61-3-8344-6914, mailto: hscheule@unimelb.edu.au 1 The authors would like to thank Bruce Arnold, Louis Ederington, Katrina Ellis, Bruce Grundy, Spencer Martin and Greg Schwann for valuable suggestions. The authors would also like to thank the participants of ﬁnancial seminars at the Leibniz University Hannover, Hong Kong Institute for Monetary Research, Monash University, The University of Melbourne and the 2011 Eastern Finance Association Conference. The support of the Australian Centre for Financial Studies, the Hong Kong Institute for Monetary Research, and the Thyssen Krupp foundation is gratefully acknowledged. Preprint submitted to Working Paper 1 May 2011 Forecasting Mortgage Securitization Risk under Systematic Risk and Parameter Uncertainty Abstract The Global Financial Crisis (GFC) exposed ﬁnancial institutions to severe unex- pected losses in relation to mortgage securitizations and derivatives. This paper develops a simple model for default and correlation of rated mortgage-backed secu- rities and home equity loan securitizations. The analysis of an extensive ratings and impairment database ﬁnds that risk models such as ratings do not reﬂect systematic risk and are exposed to a large degree to parameter uncertainty. An out-of-sample forecasting exercise of the ﬁnancial crisis shows that a simple approach addressing both issues would have been able to produce ranges for risk measures which would have covered realized losses. This may explain some of the ‘surprise’ of ﬁnancial markets in relation to realized losses. JEL classiﬁcation: G20; G28; C51 Keywords: Economic capital; Global Financial Crisis; Home Equity Loan Security; Mortgage-backed Security; Parameter Uncertainty; Rating; Securitization; Systematic Risk; Value-at-Risk 2 1 Introduction Shortcomings of securitization rating models applied by credit rating agencies (CRAs) were identiﬁed as a source of the Global Financial Crisis (GFC) (see e.g., Hellwig 2008, Hull 2009, Crouhy et al. 2008). Financial markets were surprised by high levels of impairment rates and massive downgrades of seemingly high quality (e.g., AAA-rated) mortgage-backed securities in 2007 and 2008. Figure 1 shows a representative example, which compares the impairment rates for Baa-rated mortgage-backed securities (MBS) with Baa-rated home equity loan securities (HEL). Both MBS and HEL are securitizations of real-estate collateralized loan portfolios. 2 Impairment rates for Baa-rated MBS and Baa-rated HEL are well below 10% before the GFC and peak at 29.2% (MBS) and 46.0% (HEL), respectively, during the GFC. 3 [insert Figure 1 here] This paper looks at two properties of ratings-based risk models: (i) the large exposure of securitized tranches to systematic risk, and (ii) the instability or uncertainty of model pa- rameters. We show that much of the high impairment rates during the GFC would have been not been as surprising if CRA rating models or the interpretations of such by ﬁnancial markets would have been accounted for systematic risk and parameter uncertainty. This paper relates to several streams in the literature. One stream measures the inherent actuarial credit risk (also known as physical risk) of the underlying asset portfolio of a secu- ritization transaction. The main goal is to develop approaches for modeling and forecasting the distribution of future credit losses based on individual risk parameters, such as the de- fault probability. The parameters are aggregated to a portfolio risk distribution. Important approaches which address the default probability are due to Merton (1974), Leland (1994), 2 MBS are collateralized by prime mortgages and HEL securities are mostly collateralized by sub- prime mortgages. 3 The analysis of other rating classes shows similar time-series characteristics, namely a jump in the impairment rate during the GFC. 3 Jarrow & Turnbull (1995), Longstaﬀ & Schwartz (1995), Madan & Unal (1995), Leland & Toft (1996), Jarrow et al. (1997), Duﬃe & Singleton (1999), Shumway (2001), Carey & Hrycay (2001), Crouhy et al. (2001), Koopman et al. (2005), McNeil & Wendin (2007) and Duﬃe et al. (2007). In addition, Dietsch & Petey (2004) and McNeil & Wendin (2007) model the correlations between default events and Lee et al. (2011) between the underlying asset value process variables. Carey (1998), Acharya et al. (2007), Pan & Singleton (2008), Qi & a Yang (2009), Grunert & Weber (2009) and Bruche & Gonz´lez-Aguado (2010) develop eco- nomically motivated empirical models for recoveries using explanatory co-variables. Altman et al. (2005) model correlations between default events and loss rates given default. Another stream uses market prices of credit derivatives, which are structurally similar to securitizations and develops risk-neutral pricing models. Prominent approaches are due to Li (2000), Hull & White (2004) and Longstaﬀ & Rajan (2008). A third stream of literature deals with rating issues before and during the GFC. Benmelech & Dlugosz (2009) show empirically that rating inﬂation was an issue in the GFC and they argue that one of the causes of the crisis was overconﬁdence in statistical models. The authors use rating migration statistics and analyze up- and downgrades during the crisis. Ashcraft et al. (2009) ﬁnd that CRA ratings for mortgage-backed securities provide useful information for investors, show signiﬁcant time variation and become less conservative prior to the GFC. Griﬃn & Tang (2009) compare CRA model methodologies with CRA ratings for collateralized debt obligations and ﬁnd that rating models are more accurate than the actual o ratings. R¨sch & Scheule (2011) compare capital adequacy rules with risk characteristics of securitizations and ﬁnd capital arbitrage opportunities. The fourth stream analyzes the impact of systematic risk and parameter uncertainty. With o regard to systematic risk, Loeﬄer (2004) and R¨sch (2005) ﬁnd that the default prediction power of ratings for bonds is low due to the ‘though-the-cycle’- nature of CRA rating systems which implies that CRAs aim to rate by considering borrower speciﬁc information and not 4 macroeconomic information. Parameter instability and uncertainty has been addressed for market risk by Jorion (1996) and more recently by Tarashev & Zhu (2008) for credit risk. It is an important issue in secutitization that time series information for parameter estimation is limited due to the recent origination. 4 This view is supported for securitizations by Coval et al. (2009) who show that variations of the pool default correlation may have a substantial impact on the risk of the tranches and Heitﬁeld (2009) who provides a simulation study, which shows the impact of estimation errors in pool correlations on the risk measures and ratings of tranches. Next to the above analytical and simulation exercises, this paper complements these contri- butions by providing an empirical analysis of systematic risk and parameter uncertainty for securitizations with data before and during the ﬁnancial crisis. The paper takes models developed in the ﬁrst and second stream as a starting point and extends the third and fourth stream of the literature by providing empirical evidence for the aforementioned model deﬁciencies during the GFC. The paper develops a simple model for securitization impairment risk based on a standard Merton (1974)-type approach, which allows for exposure of the tranches to a systematic ‘super-factor’ which represents the econ- omy. This model is empirically calibrated to a comprehensive panel data set of ratings and impairments of securitizations. The paper shows the magnitudes of systematic risk exposure and parameter uncertainty. An out-of-sample forecasting analysis of the ﬁnancial crisis shows that a simple approach addressing both issues would have been able to produce ranges for risk measures which cover realized losses. Systematic risk and parameter uncertainty may have not been included in impairment risk measures prior to the GFC. If they had been, the high impairment rates would not have been as ‘surprising’ as observed. The rest of the paper is organized as follows. Section 2 provides the data generation and 4 The same argument holds more generally for credit portfolio risk modeling due to the recent start of data collection. 5 description. Section 3 develops a model for the default probability and portfolio loss for mortgage asset pools and securitizations thereof and shows the empirical results. Section 4 summarizes the main ﬁndings. 2 The Data The paper analyzes a comprehensive panel data set of Moody’s-rated US mortgage secu- ritization (MBS and HEL) during the years 1997 to 2008. The data contains ratings and loss events for 164,002 MBS and HEL securitizations. Loss events are traditionally called impairment events for securitizations. An impairment event is deﬁned as (compare Moody’s Investors Service 2008): “[...] one of two categories, principal impairments and interest impairments. Principal impairments include securities that have suﬀered principal write-downs or principal losses at maturity and securities that have been downgraded to Ca/C, even if they have not yet experienced an interest shortfall or principal write-down. Interest impairments, or interest- impaired securities, include securities that are not principal impaired and have experienced only interest shortfalls.” Table 1 shows the number of observations and default rate per rating category for mortgage- backed securities (Panel A) and home equity loan securitizations (Panel B). The number of observed tranches increases over time, which reﬂects the growth of these ﬁnancial instruments over recent years. The impairment rate increases during the GFC (2007 and 2008) and more generally from rating grades Aaa-A (Aaa, Aa and A) to Baa to Ba to B to Caa. Generally speaking, impairment rates for given rating categories are higher for HELs than for MBSs. HELs include to a large degree sub-prime mortgage loans and the impairment risk increased to a larger degree than the one for MBSs. 6 [insert Table 1 here] For securitization ratings, no empirical evidence that ratings for diﬀerent CRAs share the same features has been presented before. 5 Therefore, we hand-collect the initial ratings of 1,000 randomly selected tranches and assign numbers from 1 (rating Aaa for Moody’s and rating AAA for Standard & Poor’s and Fitch respectively) to 21 (rating C). Of the 1,000 tranches rated by Moody’s, 680 are rated by Standard & Poor’s and 356 are rated by Fitch. We ﬁnd extremely high Spearman correlations 6 coeﬃcients in excess of 90%: between Moody’s and Standard & Poor’s: 0.9339, between Moody’s and Fitch: 0.9584 and between Standard & Poor’s and Fitch: 0.9855. Moody’s and Standard & Poor’s diﬀer in 88 of 680 cases. Moody’s and Fitch diﬀer in 49 of 356 cases and Standard & Poor’s and Fitch diﬀer in 11 of 163 cases. This implies that the empirical likelihood of a rating deviation is between 6.7% and 13.8%. Please note that most of ratings’ diﬀerences relate to a single notch such as a rating ‘A1’ by Moody’s and ‘A’ by Standard & Poor’s. These ﬁndings suggest that results based on major CRA may be generalized to others. 3 Empirical Analysis 3.1 Model Framework Securitizations are investments in special purpose ﬁrms, which invest in a portfolio of assets. The repayment of these investments is linked to the cash ﬂows of the underlying asset portfolios. The asset portfolio (also known as pool) of a deal generally consists of ﬁnancial 5 Guettler & Wahrenburg (2007) ﬁnd that bond ratings by Moody’s and Standard & Poor’s are highly correlated. Interviews with employees of the three CRAs support the conjecture that the information content is similar for ratings of the three major CRAs. In addition, Livingston et al. (2010) ﬁnd that the impact of Moody’s bond ratings on market reactions is slightly stronger com- pared to Standard & Poor’s and supports the use of Moody’s ratings. 6 We chose to report this measure for the relationship as ratings are ordinal in nature. We obtain similar results for Bravais-Pearson correlation coeﬃcients. 7 assets (e.g., generally loans) that are subject to ﬁnancial risk (e.g., generally credit risk). Therefore, investments in securitizations cover – within legal maturities – losses to the asset portfolio in excess of a retention (also known as attachment or subordination level) and up to a limit (also known as detachment level). The paper refers to the entire transaction as ‘deal’ and the individual investment segment as ‘tranche’. In other words, one deal may consist of one or more tranches of various seniority levels. For modeling the asset pool risk and the impairment risk of securitized tranches thereof we follow Vasicek (1987, 1991), Gordy (2000), Li (2000), Gordy (2003) who develop a latent factor credit risk model consistent with Merton (1974). It is assumed that the latent returns in the asset pool are driven by systematic and idiosyncratic and therefore diversiﬁable factors and that the asset portfolio is inﬁnitely granular (see Gordy 2000, 2003). This implies that idiosyncratic risk is fully diversiﬁed away. The model is also known as asymptotic single risk factor (ASRF) model or Gaussian copula model and is a ‘market standard’ for quoting and pricing CDOs, see Li (2000). These models have also found their recognition in the supervisory rules for determining regulatory capital of banks (i.e., Basel II and Basel III). The default rate of pool i in time period t (i = 1, ..., I; t = 1, ..., T ) is then modeled as √ cit − ρit Xit Pit = Φ √ (1) 1 − ρit where Xit is a time-speciﬁc systematic risk factor which aﬀects all assets in the pool jointly. √ ρit is the exposure of the asset returns in the pool to this factor. Φ(·) is the cumulative distribution function (CDF) of the standard normal distribution. cit = Φ−1 (πit ) is the default threshold, where πit is the probability of default (PD) and Φ−1 (·) is the inverse of the standard normal CDF. The parameter ρit is also known as ‘asset correlation’. 8 The density f (·) and CDF F (·) of the default rate Pit in pool (or deal) i are then given by √ 1 − ρit 1 −1 1 f (pit ) = √ · exp (Φ (pit ))2 − (cit − 1 − ρit · Φ−1 (pit ))2 (2) ρit 2 2ρit √ 1 − ρit Φ−1 (pit ) − Φ−1 (πit ) F (pit ) = Φ √ (3) ρit (4) Impairment of a tranche j occurs if the pool default rate is higher than the attachment level ALijt of the tranche, i.e., Pit > ALijt . The tranche impairment probability is then given as √ 1 − ρit Φ−1 (ALijt ) − Φ−1 (πit ) P (Dijt = 1) = 1 − Φ √ (5) ρit √ −1 Φ (πit ) − 1 − ρit Φ−1 (ALijt ) =Φ √ ρit = Φ (ηijt ) where Dijt is an indicator variable with 1 tranche j of deal i is impaired in t Dijt = (6) 0 otherwise Following Gordy & Howells (2006), we introduce an economy-wide ‘super’-factor which af- fects all pools in the economy jointly. Therefore, we model correlations across pools by decomposing the pool speciﬁc factor into 9 Xit = δit · Xt∗ + 1 − δit · Uit (7) where Xt∗ is a univariate standard normally distributed ‘super’-factor measuring the state of the economy. Uit is a pool speciﬁc factor. δit measures the strength of dependence across pools. All factors are standard normally distributed and cross-sectionally as well as serially independent. We assume δit = δ for all pools for eﬃciency. Then the conditional tranche impairment probability can be stated as a function of the systematic factor by √ √ √ Φ−1 (πit ) − 1 − ρit Φ−1 (ALijt ) − ρit δXt∗ P (Dijt = 1|Xt∗ ) =Φ √ √ (8) ρit 1 − δ √ = Φ ηijt / 1 − δ + b · Xt∗ (9) √ √ where b = − δ/ 1 − δ is the transformed exposure to the ‘super-factor’. √ The expression ηijt / 1 − δ may be modeled by observable tranche characteristics such that √ ηijt / 1 − δ = β xijt , where xijt are observable variables and β is a vector of parameters. The model can then be stated in terms of a mixed eﬀects probit regression (with ﬁxed eﬀects xijt and random eﬀects Xt∗ ) or ‘frailty’ model (see Duﬃe et al. 2009) as P (Dijt = 1|Xt∗ ) = Φ (β xijt + b · Xt∗ ) (10) It is obvious to see that the higher the degree to which tranches are exposed to the common economy-wide factor, the higher the standard deviation b is and the higher the deviations of the realized tranche impairment probability from the expected probability of Equation (5) are. 10 The dispersion parameter b of the random eﬀect model can be interpreted in a similar fashion after reparameterization as ‘asset correlation’. 7 √ β xijt + δ · Xt∗ P (Dijt = 1|Xt∗ ) = Φ √ (11) 1−δ b2 √ where δ = 1+b2 and β = β · 1 − δ. In other words, δ can be interpreted as the correlation between two asset returns which trigger a tranche impairment when crossing the thresholds β xijt . The parameters of the random eﬀect model are estimated by the Maximum Likelihood o method as outlined in Hamerle & R¨sch (2006) and McNeil & Wendin (2007) for common credit portfolios. We estimate the model both using several tranches per deal and using one tranche per deal as a robustness check. 8 The results are comparable. Table 2 shows the estimation results for MBS and HEL securitizations as well as the following sample periods: • All: whole sample period 1997-2008; • Pre 2008: restricted sample period 1997-2007; and • Pre 2007: restricted sample period 1997-2006. [insert Table 2 here] We include the credit ratings of the tranches as independent variables. The coeﬃcients for the ratings are increasing with decreasing rating quality. This is in line with our expectation as well as the descriptive analysis as a lower credit quality should imply a higher default probability. For all risk segments, the lower the credit quality, the higher the estimated 7 Asset correlations are an important parameter in the Internal Ratings-based approach in Basel II and Basel III. 8 This approach applies a bootstrap method to use the data most eﬃcient and is described in Section 3.3. 11 default risk of a tranche. The coeﬃcients for the unobservable macroeconomic eﬀect are statistically signiﬁcantly diﬀerent from zero given the credit ratings in all models. 9 We ﬁnd larger diﬀerences for the comparison of the estimates for the whole sample period with the pre-crisis periods (pre 2007 and pre 2008, respectively). The coeﬃcients for the credit ratings change as well as the exposure to the economic factor diﬀer for MBS and HEL. The exposure to the economic factor are higher for HEL than for MBS. These coeﬃcients increase with the inclusion of a higher degree of information on the GFC. This is due to the large increase in defaults during the crisis, which is not captured in the data prior to the GFC. The previous model assumes that exposures to the economy are homogeneous across the rating grades. In order to allow for heterogenous exposures, the models are estimated for each rating grade separately. The results are shown in Table 3. [insert Table 3 here] The exposure of the macroeconomic factor varies greatly between rating grades. The expo- sures are generally higher for HEL than for MBS. In addition, we ﬁnd that including the data from the ﬁnancial crisis generally increases the coeﬃcients except for investment grade rated MBS. Note that the estimated standard errors are high for the macroeconomic exposures. This reﬂects the high degree of estimation uncertainty, which is induced by the relative short time series. Predictions Using Point Estimates for the Parameters The model enables the prediction of the impairment probability associated with each rating grade after the parameter estimates δ and β are obtained. Due to the simple reparameteriza- tion and the analogy to the asset value model, the predicted density of the tranche j default 9 We have tested ﬁxed year eﬀects in an unreported study. In many years the estimates for the year dummies are signiﬁcantly diﬀerent from zero and positive during economic downturns such as the recent ﬁnancial crisis. This shows that time-speciﬁc (economic) inﬂuences are not able to be wholly explained by the rating. Details are available from the authors upon request. 12 rate pijt is given by 1−δ 1 −1 1 f (pijt ) = · exp (Φ (pijt ))2 − (β xijt − 1 − δ · Φ−1 (pijt ))2 (12) δ 2 2δ which is conditional on the parameter estimates δ and β and the observable variables con- tained in xijt . For instance, given the parameter estimates of the pre 2007 HEL model the densities for the various rating grades are shown in Figure 2 and Figure 3. In this setting, considerable uncertainty is induced by the exposure to the macroeconomic risk factor. [insert Figure 2 here] [insert Figure 3 here] The predicted distribution function is given by −1 1 − δΦ (pijt ) − β xijt F (pijt ) = P (Pijt < pijt ) = Φ (13) δ and the α-percentile (which can be interpreted as the value-at-risk) can be calculated as −1 β xijt + δΦ (α) qα = Φ (14) 1−δ For both pool segments HEL and MBS and the pre 2007 and pre 2008 models, we check the likelihood of the realized pool default rate in the subsequent year (i.e., out-of sample) under the respective model. In other words, we use the estimates of the models using data up to 13 2006 and calculate the probability of observing the realized default rate (or a higher one) in year 2007 by inserting the realized default rate into Equation (13). We proceed analogously for the pre 2008 models. This methodology provides an assessment of how well the model performs out-of-time using rating data prior to the GFC. In other words, we simulate an investor who (i) analyzes the impairment rates of HEL and MBS pools given the information up to 2006 and 2007 respectively and (ii) calculates the distribution of the potential default rates using historical information. Such an investor ac- knowledges that rating agencies do not include all macroeconomic information which may aﬀect the asset pool tranches via an unexpected shock. The investor calculates the eco- nomic capital (ECAP) as the value-at-risk according to Equation (14). We exemplarily use a conﬁdence level of 99.98%. 10 Table 5 shows the results for the model with homogenous systematic exposure for all rating grades. Table 6 contains the results for the rating grade speciﬁc exposures. The realized impairment rate exceeds the economic capital for most rating grades of 2007 and all grades of 2008 for MBS as well as for HEL. This implies that leveraged investors 11 applying the prediction model to determine the minimum capital would have not had a suﬃcient level of capital to cover the losses. In 2008, the level of capital shortfall is extremely large and the implied probability of occurrence of the realized impairment rate is close to zero. This is a result, which conﬁrms our expectation, as the GFC exposed many ﬁnancial institutions to losses in relation to securitizations. [insert Table 5 here] 10 Deutsche Bank for instance reports economic capital on a 99.98% conﬁdence level, see Deutsche Bank (2009). This level is conﬁrmed by Hull (2010), who oﬀers a conﬁdence level of 99.97% as a reference value banks often use for internal economic capital calculations. 11 This includes banks and insurance ﬁrms which are subject to regulatory capital and reserve requirements. 14 [insert Table 6 here] 3.2 Parameter Uncertainty We have used the Maximum-Likelihood point estimates for forecasting in the previous chap- ter. However, Table 3 shows that the standard errors for the estimates may be substantial. This is particularly true for the random eﬀect parameter. For example for AAA-A rated MBS securities in the pre 2007 period the coeﬃcient is 0.4512 and the its standard devia- tion estimate is 0.3151 which is approximately 67% of its size. This means, an investor who associates coeﬃcients (e.g., macroeconomic exposure) with a securitized tranche, can not be sure that the estimate of the coeﬃcient is correct. This problem is particularly pronounced for short time series (see e.g., Gordy & Heitﬁeld 2000). 12 In the following we predict the impairment probability density associated with each rat- ing grade under parameter uncertainty. In the credit risk area estimation errors have been addressed by Loeﬄer (2003), Tarashev & Zhu (2008) and Heitﬁeld (2009) by Monte-Carlo o simulation studies. We follow Hamerle & R¨sch (2005) who suggest an approach which sim- ulates distributions for the value-at-risk similarly to Jorion (1996) and therefore takes the parameter uncertainty into account. The estimated covariance of the parameter estimates is deﬁned as Cov ψ = Σ (15) 12Gordy & Heitﬁeld (2000) also show that the Maximum-Likelihood estimator is downwards biased in small samples which means that an investor underestimates the coeﬃcient on average. The underestimation for our time-series length of 10 or 11 years is about 20%. 15 where ψ is the vector containing all parameter estimates. We randomly draw sample real- izations for the parameter estimates using this covariance matrix according to ψ = ψ · cadj + Σ 0.5 · (16) where cadj is a correction factor for the bias adjustment (we set cadj to 1.2 which relates to an underestimation of approximately 20%), is a standard normally distributed random variable, and Σ0.5 is the Cholesky Decomposition such that Σ 0.5 · Σ0.5 = Σ. We calculate the value-at-risk (i.e., the economic capital) under parameter uncertainty given each sample of random realizations for the parameter estimates according to β xijt + δΦ−1 (α) qα = Φ (17) 1−δ We draw 100,000 random samples for each rating grade and asset pool and obtain a dis- tribution of the 99.98%-economic capital for HEL and MBS under parameter uncertainty for each rating grade. Figure 4 and Figure 5 show the economic capital (z-axis) for each of the 100,000 randomly simulated settings for the two parameters (β xijt on the x axis, and b on the y-axis) for years 2007 and 2008 respectively. Figure 6 and Figure 7 summarize the economic capital scatters in frequency distributions. The vertical grey lines show the actual default rates in that segment in the respective year. The ﬁgures show that the empirical parameter uncertainty for both MBS and HEL as well as for both years is high. Particularly in 2008, where the default rates for HEL are very high (more than 13% for the investment grade rating classes and almost 95% in the Caa rating class), we see that the impact of estimation error on economic capital leads to distributions 16 which are centered around the realized default rates. 13 A ﬁnancial market participant, who had been aware of the high degree of systematic risk and parameter uncertainty before the ﬁnancial crisis would have calculated capital buﬀers, which would have covered much of the eﬀects of the ‘surprisingly’ high impairment rates of securitizations and losses thereof. [insert Figure 4 here] [insert Figure 5 here] [insert Figure 6 here] [insert Figure 7 here] 3.3 Robustness Check: Tranche Dependencies The empirical probit regression model assumes independence of the tranches. However, the impairment events may exhibit dependence between the tranches of the same deal. 14 Therefore, a three-step bootstrap technique is applied. In the ﬁrst step, the models from Table 2 are estimated for one randomly drawn tranche per deal (i.e., the random selection is stratiﬁed per deal). In the second step, Step 1 is repeated 200 times. In Step 3, the average parameter estimates and their standard deviations are calculated. Table 4 shows these estimates for Table 2. The values are close to the estimates we obtain in Table 2 for MBS and HEL despite the larger standard errors. We apply the same techniques to other outputs of Table 2 and Table 3. The results are consistent in all instances. [insert Table 4 here] 13 The only segment where this is not the case is rating class B of MBS in 2008. Please note that the number and the volume of securities within this segment is rather low. 14 The authors would like to thank our seminar participants for raising this concern. 17 4 Summary We identify and empirically assess two related issues associated with the measurement of risk in relation to securitizations: parameter uncertainty and systematic risk. Our main ﬁndings are as follows: Firstly, our empirical analysis supports the contributions in relation to parameter uncertainty by Coval et al. (2009) and Heitﬁeld (2009), who show that variations of the pool default correlation can have a substantial impact on the risk of the tranches. Secondly, we show that credit ratings do not reﬂect the systematic risk appropriately. As a result, securitized tranches – with low default risk in booms – may experience much higher impairment rates in an economic downturn. 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(2011), ‘Capital incentives and capital adequacy for securitizations’, forthcoming Journal of Banking and Finance . Shumway, T. (2001), ‘Forecasting bankruptcy more accurately: A simple hazard-rate model’, Journal of Business 74, 101–124. Tarashev, N. & Zhu, H. (2008), ‘Speciﬁcation and calibration errors in measures of portfo- 22 lio credit risk: The case of the ASRF model’, International Journal of Central Banking pp. 129–173. Vasicek, O. (1987), Probability of loss on loan portfolio, Working paper, KMV Corporation. Vasicek, O. (1991), Limiting loan loss probability distribution, Working paper, KMV Cor- poration. 23 Tables 24 Table 1 Total number of observations and impairment rates, MBS and HEL, 1997-2008 This table shows the number of observations (NO) and impairment rate (IR) per rating category for mortgage-backed securities (MBS, Panel A) and home equity loan securitizations (HEL, Panel B) from 1997 to 2008. The number of observed tranches increases over time which reﬂects the growth of these ﬁnancial instruments during recent years. The impairment rate increases during the GFC (2007 and 2008) and more generally from rating grades Aaa-A (Aaa, Aa and A) to Baa to Ba to B to Caa. Generally speaking, impairment rates for given rating categories are higher for HELs than for MBSs. HELs include to a large degree sub-prime mortgage loans and the impairment risk increased to a larger degree than the one of MBSs. Panel A: MBS Year All Grades Aaa-A Baa Ba B Caa-C NO IR NO IR NO IR NO IR NO IR NO IR 1997 7,377 0.0009 6,820 0.0000 304 0.0033 167 0.0180 86 0.0349 1998 7,715 0.0003 7,051 0.0000 369 0.0000 180 0.0056 114 0.0088 1 0.0000 1999 7,458 0.0008 6,629 0.0000 454 0.0000 213 0.0047 151 0.0133 11 0.2727 2000 7,361 0.0007 6,356 0.0000 535 0.0037 249 0.0080 204 0.0049 17 0.0000 2001 7,632 0.0012 6,474 0.0000 602 0.0017 281 0.0036 250 0.0240 25 0.0400 2002 9,131 0.0022 7,574 0.0000 826 0.0048 384 0.0260 321 0.0187 26 0.0000 2003 10,557 0.0021 8,434 0.0001 1,103 0.0036 540 0.0074 440 0.0136 40 0.1750 2004 10,290 0.0020 7,928 0.0004 1,156 0.0009 643 0.0000 513 0.0195 50 0.1400 2005 12,857 0.0017 9,987 0.0000 1,374 0.0007 813 0.0049 608 0.0115 75 0.1333 2006 20,229 0.0011 16,363 0.0000 2,025 0.0005 1,036 0.0010 710 0.0099 95 0.1474 2007 28,859 0.0033 23,677 0.0001 2,937 0.0170 1,331 0.0188 822 0.0146 92 0.0544 2008 34,536 0.0839 28,051 0.0315 3,324 0.2924 1,743 0.3138 1,197 0.2924 221 0.6516 All 164,002 0.0191 135,344 0.0027 15,009 0.0274 7,580 0.0343 5,416 0.0388 653 0.1468 Panel B: HEL Year All Grades Aaa-A Baa Ba B Caa-C NO IR NO IR NO IR NO IR NO IR NO IR 1997 1,630 0.0141 1,528 0.0000 53 0.0189 30 0.3667 19 0.5790 1998 2,401 0.0079 2,243 0.0013 111 0.0631 31 0.2258 16 0.1250 1999 2,982 0.0097 2,735 0.0011 154 0.0844 62 0.1129 29 0.2069 2 0.0000 2000 3,297 0.0049 2,989 0.0000 216 0.0139 49 0.0612 30 0.1333 13 0.4615 2001 3,579 0.0034 3,224 0.0003 247 0.0122 71 0.0423 29 0.1724 8 0.0000 2002 3,903 0.0036 3,422 0.0000 348 0.0115 93 0.0215 27 0.1111 13 0.3846 2003 4,462 0.0058 3,787 0.0000 543 0.0184 95 0.0947 25 0.2000 12 0.1667 2004 5,493 0.0020 4,416 0.0002 932 0.0043 104 0.0289 27 0.0741 14 0.0714 2005 7,999 0.0021 6,109 0.0000 1,633 0.0018 204 0.0147 40 0.2250 13 0.1539 2006 12,549 0.0016 9,183 0.0000 2,716 0.0022 596 0.0084 43 0.1628 11 0.1818 2007 18,339 0.0553 13,059 0.0074 4,017 0.1061 1,142 0.3853 95 0.3263 26 0.7692 2008 19,752 0.2900 13,503 0.1326 3,303 0.4593 1,455 0.7464 1,095 0.8758 396 0.9495 All 86,386 0.0802 66,198 0.0119 14,273 0.0663 3,932 0.1757 1,475 0.2660 508 0.3139 25 Table 2 Parameter estimates of random eﬀects models, MBS and HEL This table shows parameter estimates from the random eﬀects probit model. Standard errors are below each estimate. The signiﬁcance is indicated as follows: ***: signiﬁcant at 1%, **: signiﬁcant at 5%, *: signiﬁcant at 10%. AIC is the Akaike Information Criterion. The coeﬃcients for the unobservable random eﬀect are statistically signiﬁcantly diﬀerent from zero in all models. After including credit ratings, the coeﬃcients are higher. This underlines that ratings do not properly account for macroeconomic eﬀects. The comparison of the estimates for the whole sample period with the pre-crisis periods (pre 2008 and pre 2007) reveals large diﬀerences. The coeﬃcients for the credit ratings change both for MBS and HEL and the estimates for the exposure to the economic factor are higher for HEL. These coeﬃcients increase with the inclusion of a higher degree of information on the GFC. This is due to the large increase in defaults during the crisis which is not captured by using data prior to the GFC. MBS HEL all pre-2008 pre-2007 all pre-2008 pre-2007 Intercept -3.6646*** -3.7164*** -3.9115*** -3.0967*** -3.4228*** -3.6745*** Std error 0.1713 0.0994 0.1270 0.2207 0.1607 0.1746 Baa 1.2830*** 1.2929*** 0.9834*** 1.0628*** 1.2339*** 1.4815*** Std error 0.0255 0.1046 0.1464 0.0213 0.0421 0.1216 Ba 1.3732*** 1.4816*** 1.3998*** 1.8955*** 2.1133*** 2.1021*** Std error 0.0312 0.1082 0.1400 0.0284 0.0477 0.1276 B 1.4019*** 1.6938*** 1.7304*** 2.3011*** 2.4644*** 2.8351*** Std error 0.0357 0.1079 0.1347 0.0432 0.0851 0.1380 Caa 2.4143*** 2.6824*** 2.7825*** 2.7984*** 2.9813*** 3.0740*** Std error 0.0646 0.1259 0.1522 0.0849 0.1400 0.1932 b 0.5782*** 0.3166*** 0.1075 0.7564*** 0.5102*** 0.4382*** Std error 0.1197 0.0754 0.0481 0.1555 0.1121 0.1051 Obs 164,002 129,466 100,607 86,386 66,634 48,295 AIC 18,281 2,460 1,414 24,864 7,054 1,477 26 Table 3 Parameter estimates of random eﬀects models, MBS and HEL, per rating category This table shows parameter estimates from the random eﬀects probit model per rating category. Standard errors are in parentheses. The signiﬁcance is indicated as follows: ***: signiﬁcant at 1%, **: signiﬁcant at 5%, *: signiﬁcant at 10%. AIC is the Akaike Information Criterion. The coeﬃcients for the unobservable macroeconomic eﬀect are statistically signiﬁcantly diﬀerent from zero in all models. We see large diﬀerences, if we compare the estimates for the whole sample period with the pre-crisis periods (from 2006 and until 2007). The exposure of the macroeconomic factor varies greatly between the rating grades. The exposures are generally higher for HEL and we see that including the data from the ﬁnancial crisis generally increases the coeﬃcients except for investment grade rated MBS. Note also that the estimated standard errors are very high for the macroeconomic exposures. MBS HEL Panel A: Aaa-A all pre-2008 pre-2007 all pre-2008 pre-2007 Intercept -4.4068*** -4.0027*** -4.2339*** -3.6863*** -3.7037*** -3.7291*** std error 0.5923 0.2415 0.3988 0.4339 0.2954 0.2386 b 1.2058** 0.3021 0.4512 1.2031*** 0.6863** 0.4286* std error 0.5033 0.2163 0.3151 0.3764 0.2560 0.2195 Obs 135,344 107,293 83,616 66,198 52,695 39,636 AIC 8,015 151 88 11,891 1,306 151 Panel B: Baa Intercept -2.7711*** -2.8825*** -2.9434*** -1.9722*** -2.1468*** -2.2495*** std error 0.2617 0.1434 0.1022 0.2305 0.1726 0.1610 b 0.8301*** 0.3629*** 0.1537 0.7753*** 0.5381*** 0.4663*** std error 0.1954 0.1080 0.1228 0.1621 0.1232 0.1181 Obs 15,009 11,685 8,748 14,273 10,970 6,953 AIC 4,780 742 224 7,872 3,299 573 Panel C: Ba Intercept -2.3793*** -2.5037*** -2.5628*** -1.2555*** -1.4356*** -1.5636*** std error 0.2242 0.1282 0.1338 0.2626 0.2156 0.1989 b 0.7241*** 0.3380** 0.3254** 0.8833*** 0.6786*** 0.5827*** std error 0.1663 0.1079 0.1153 0.1865 0.1566 0.1506 Obs 7,580 5,837 4,506 3,932 2,477 1,335 AIC 2,763 575 322 3,585 1,924 393 Panel D: B Intercept -2.0515*** -2.1846*** -2.1855*** -0.6768*** -0.8373*** -0.8958*** std error 0.1585 0.0501 0.1338 0.2155 0.1234 0.1271 b 0.5104*** 0.0000 0.0000 0.6953*** 0.3047** 0.2844* std error 0.1108 0.0784 0.1083 0.1527 0.1186 0.1356 Obs 5,416 4,219 3,397 1,475 380 285 AIC 2,120 642 517 1,242 402 278 Panel E: Caa Intercept -1.2087*** -1.2825*** -1.1575*** -0.5364*** -0.7382*** -0.8808*** std error 0.2610 0.1333 0.0874 0.3870 0.2988 0.2193 b 0.7322*** 0.2437 0.0000 1.0807*** 0.7141** 0.3517 std error 0.2127 0.1753 0.2520 0.3006 0.2473 0.2440 Obs 653 432 340 508 112 86 AIC 599 300 258 296 127 91 27 Table 4 Parameter estimates of random eﬀects models, MBS and HEL, per rating category; bootstrap methodology This table shows averages of the parameter estimates from the random eﬀects probit model using a bootstrap methodology in Column 2 and Column 4. The empirical standard deviations are given in Column 3 and Column 5. The results are consistent to the ones presented in Table 2 (MBS and HEL for complete data set). MBS HEL Average Std. dev. Average Std. dev. Intercept -3.7302 0.2159 -3.0752 0.0649 Baa 1.2318 0.0853 1.0597 0.0629 Ba 1.5144 0.0850 1.7494 0.0774 B 1.4987 0.1202 2.1394 0.0884 Caa 2.0270 0.1508 2.5762 0.1191 b 0.6416 0.0518 0.7428 0.0628 28 Table 5 Model out-of-time performance, homogenous systematic exposure, per rating category This table shows the results for the model with homogenous macroeconomic exposure for all grades (i.e., models shown in Table 2). The realized impairment rate exceeds the economic capital for most rating grades of 2007 (Panel A) and all grades of 2008 (Panel B) for both for MBS as well as for HEL. In 2008, the level of capital shortfall is very high and the implied probability of occurrence of the realized impairment rate is close to zero. Panel A: 2007 MBS HEL Rating Implied Correlation PD estimate Impairment Rate 99.98 ECAP Prob. Implied Correlation PD estimate Impairment Rate 99.98 ECAP Prob. Aaa-A 0.0114 0.0001 0.0001 0.0002 0.0094 0.1611 0.0004 0.0074 0.0169 0.0024 Baa 0.0114 0.0018 0.0170 0.0054 0.0000 0.1611 0.0223 0.1060 0.2605 0.0155 Ba 0.0114 0.0063 0.0188 0.0165 0.0000 0.1611 0.0749 0.3853 0.4915 0.0017 B 0.0114 0.0151 0.0146 0.0359 0.4988 0.1611 0.2210 0.3263 0.7617 0.1872 Caa-C 0.0114 0.1308 0.0543 0.2271 1.0000 0.1611 0.2912 0.7692 0.8291 0.0011 Panel B: 2008 29 MBS HEL Rating Implied Correlation PD estimate Impairment Rate 99.98 ECAP Prob. Implied Correlation PD estimate Impairment Rate 99.98 ECAP Prob. Aaa-A 0.0911 0.0002 0.0315 0.0047 0.0000 0.2065 0.0011 0.1326 0.0530 0.0000 Baa 0.0911 0.0104 0.2924 0.0963 0.0000 0.2065 0.0256 0.4593 0.3510 0.0000 Ba 0.0911 0.0166 0.3138 0.1326 0.0000 0.2065 0.1217 0.7464 0.6903 0.0001 B 0.0911 0.0269 0.2924 0.1835 0.0000 0.2065 0.1966 0.8758 0.8017 0.0000 Caa-C 0.0911 0.1621 0.6516 0.5345 0.0000 0.2065 0.3471 0.9495 0.9138 0.0000 Table 6 Model out-of-time performance, heterogeneous systematic exposure, per rating category This table contains the results for the rating grade speciﬁc exposures (i.e., models shown in Table 3). The realized impairment rate exceeds the economic capital for most rating grades of 2007 (Panel A) and all grades of 2008 (Panel B) for both for MBS as well as for HEL. In 2008, the level of capital shortfall is very high and the implied probability of occurrence of the realized impairment rate is close to zero. Panel A: 2007 MBS HEL Rating Implied Correlation PD estimate Impairment Rate 99.98 ECAP Prob. Implied Correlation PD estimate Impairment Rate 99.98 ECAP Prob. Aaa-A 0.1692 0.0001 0.0001 0.0042 0.1012 0.1552 0.0003 0.0074 0.0135 0.0013 Baa 0.0231 0.0018 0.0170 0.0082 0.0000 0.1786 0.0207 0.1060 0.2747 0.0159 Ba 0.0957 0.0074 0.0188 0.0791 0.0687 0.2535 0.0884 0.3853 0.6912 0.0145 B 0.0000 0.0144 0.0146 0.0144 0.0000 0.0748 0.1945 0.3263 0.5442 0.0586 Caa-C 0.0000 0.1235 0.0543 0.1235 1.0000 0.1101 0.2030 0.7692 0.6422 0.0000 Panel B: 2008 30 MBS HEL Rating Implied Correlation PD estimate Impairment Rate 99.98 ECAP Prob. Implied Correlation PD estimate Impairment Rate 99.98 ECAP Prob. Aaa-A 0.0836 0.0001 0.0315 0.0017 0.0000 0.3202 0.0011 0.1326 0.1013 0.0001 Baa 0.1163 0.0034 0.2924 0.0550 0.0000 0.2246 0.0293 0.4593 0.4045 0.0001 Ba 0.1025 0.0088 0.3138 0.0956 0.0000 0.3153 0.1174 0.7464 0.8331 0.0010 B 0.0000 0.0145 0.2924 0.0145 0.0000 0.0849 0.2116 0.8758 0.5953 0.0000 Caa-C 0.0560 0.1064 0.6516 0.3373 0.0000 0.3377 0.2740 0.9495 0.9632 0.0004 Figures 31 Fig. 1. Impairment Rates for MBS and HEL securitizations This ﬁgure compares the impairment rates for Baa-rated mortgage-backed securities (MBS) which are collateralized by prime mortgages and Baa-rated home equity loan (HEL) securities which are mostly collateralized by sub-prime mortgages. Both MBS and HEL are securitizations of real-estate linked loan portfolios. Impairment rates for Baa-rated MBS ﬂuctuate between zero and 29.2% and impairment rates for Baa-rated HEL ﬂuctuate between 0.2% and 46.0%. Values for impairment rates of MBS securitizations and HEL securitizations are shown in Table 1. 32 Fig. 2. PD predictions HEL 2007, rating classes Aaa-A and Baa This ﬁgure shows the densities for the various rating grades given the parameter estimates of the pre 2007 HEL model for the rating classes Aaa-A and Baa. Fig. 3. PD predictions HEL 2007, rating classes Ba to Caa This ﬁgure shows the densities for the various rating grades given the parameter estimates of the pre 2007 HEL model for the rating classes Ba to Caa. 33 Fig. 4. VaR Scatterplots 2007 This ﬁgure shows the 99.98%-economic capital (z-axis) for each of the 100,000 randomly simulated settings for the two parameters (β xijt on the x axis, and b on the y-axis) for 2007. 34 Fig. 5. VaR Scatterplots 2008 This ﬁgure shows the 99.98%-economic capital (z-axis) for each of the 100,000 randomly simulated settings for the two parameters (β xijt on the x axis, and b on the y-axis) for 2008. 35 Fig. 6. VaR Distributions 2007 (99.98% economic capital) This ﬁgure summarizes the economic capital scatters into frequency distributions for 2007. The vertical grey lines show the actual default rates in that segment in the respective year. 36 Fig. 7. VaR Distributions 2008 (99.98% economic capital) This ﬁgure summarizes the economic capital scatters into frequency distributions for 2008. The vertical grey lines show the actual default rates in that segment in the respective year. 37

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