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Forecasting Mortgage Securitization Risk under Systematic Risk and

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Forecasting Mortgage Securitization Risk under Systematic Risk and Powered By Docstoc
					    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.

Thirdly, given the systematic risk exposure we predict out-of-time rating-implied default
rates and take parameter uncertainty into account. We find that the high empirical default
rates of the rated tranches in the financial crisis are within expectations. This finding implies
that model properties such as securitization-induced systematic risk and parameter uncer-
tainty may complement the information provided by credit ratings. Taking systematic risk
and parameter uncertainty into account implies broad intervals for risk measures such as the
value-at-risk or economic capital. Given this knowledge, high default rates would not have
come as unexpected as they did for many market participants in the financial crisis.




                                             18
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                                           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.




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                         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.




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                         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.




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