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Thus far, the CreditRisk+ framework has been presented as an alternative structure to that found in the asset value model. However, it can nonetheless be shown that CreditRisk+ shares common conceptual and mathematical foundations with the AVM studied in Chapter 4; see, for example, Crouhy, Galia, and Mark (2000). In particular, Koyluoglu and Hickman (1998) show that both model types can be classified under a single general framework and are capable of providing similar results provided that input parameters are harmonized across the models. Gordy (2000) provides a mapping between the asset value model and the CreditRisk+ model while Gordy and Lutkebohmert (2007) present an example of how applications or tests from one framework can be applied to another by using results generated in a CreditRisk+ framework to propose amendments to the Basel II IRB framework.

Wieczerkowski (2003, pp. 45-46) summarizes results of numerous comparative studies between the CreditRisk+ and the AVM, as represented by the commercialized CreditMetrics model, see J.P. Morgan (1997), frameworks:

1. In limiting CreditMetrics to a two-state model, both it and CreditRisk+ can be considered factor models imposing conditional independence on defaults. The two models, however, differ in the modeling of the distributions of risk factors and conditional default probabilities.

2. The two models can be reformulated analogously to one another. In the case of CreditRisk+, this reformulation can generate a two-state CreditMetrics framework

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in which loss distributions are generated through Monte Carlo simulation. In the case of CreditMetrics a limited two-state model probability-generating function can be generated analogously to CreditRisk+; however, no closed form solution has been found.

3. While it is possible to match the CreditRisk+ and CreditMetrics models through factor transformations, this matching is not based on the standard parameterizations of the models.

For their part, Koyluoglu and Hickman (1998) identify three points of comparison between the frameworks – the default rate distribution, the conditional default dtribution, and the loss aggregation method – with only the first found to generate significant divergences in results; see Koyluoglu and Hickman (1998, p. 17).

In Chapter 6 we introduce a simulation-based implementation of the CreditRisk+ framework. Our comparisons of the AVM and CreditRisk+ will show that using the calibration and bridging techniques presented in Gordy (2000), it is possible to obtain comparable results for an SME portfolio under the two models. Our results emphasize the distinctiveness of the SME default rate volatilities and their importance in the calibration of portfolio credit risk models. In particular, we will show that a unitary setting to normalized SME default rate volatilities in the CreditRisk+ framework may not be suitable in an SME portfolio, especially in the presence of low correlations.

Our work will show that a unitary risk factor weighting, allowing for the calibration of the single sector risk factor volatility from internal data, as discussed in Chapter 5, provides loss distribution characteristics comparable to those obtained under the AVM. In addition, we present a calibration refinement and show that a fixed sector normalized

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standard deviation setting of 0.5 for our SME portfolio provides for parameter settings harmonized with those found in the AVM. Given the scope of our work in the Thesis, especially as it applies to a real-world portfolio of SME borrowers, this result is an interesting contribution to the literature.

Similarly to Chapter 4, correlations calibrated from internal Financing Company SME data are generally low in value. We use simulation-based implementations of the CreditRisk+ model and the calibration bridge presented in Gordy (2000) to extend the ad hoc correlation boost presented in Chapter 4 to the modeling framework used in this Chapter. This exercise, bearing similarities to that undertaken in Dietsch and Petey (2002), yields capital charges significantly higher than those obtained under the purely internal calibrations discussed above.

Finally, Chapter 6 will conclude the study of SME portfolio credit risk undertaken in this Thesis. Having studied the unique characteristics that underpin a high-risk SME loans portfolio in Chapter 2, we proceeded, in Chapter 3, with undertanding of the impact of the various assumptions on the behaviours of borrowers in an SME as found in the Basel II regulatory framework. In Chapter 4 these assumptions were tested through the estimation of SME correlations as found in our portfolio and within a comparable framework as that used in Basel II. Our results could not empirically support Basel II assertions on the relationship between SME borrower asset correlations and either probabilities of default or borrower size. In addition, our estimation of borrower asset correlations from internal Financing Company default data resulted in low correlation

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values as compared to those found in Basel II. Extending our analysis on this phenomenon, we demonstrate, empirically, the significant underestimation of portfolio credit risk that can arise from the presence of these low asset correlations in an ASRF framework. To our knowledge, this is the first demonstration of a relationship between “granularity effects” and asset correlation values in a portfolio credit risk context.

In Chapter 5 we take a broader view of the underlying credit risks in an SME portfolio by using the CreditRisk+ framework to extend our analysis from a single sector framework to one in which multiple sectors, along with the accompanying portfolio diversification benefits are modelled. Our analysis quantified the overestimation resultant from the use of a single risk factor in a portfolio credti risk model, as opposed to an internally calibrated model with multiple correlated risk factors. In addition, we quantify the underestimation that could result from the use of zero correlation.

Chapter 6 brings us full circle by comparing the AVM and CreditRisk+ framework and the resultant portfolio risk capital charges, obtained under prudentially boosted values to internally calibrated correlations, to Basel II. Specifically, Chapter 6 is divided as follows: Section 6.1 presents a single sector simulation-based CreditRisk+ model, including calibration implementations and calibrations; Section 6.2 presents a comparative analysis of AVM and CreditRisk+ results, and; Section 6.3 presents a discussion of results and conclusions. We conclude the Thesis in Section 6.4 with a review of the issues explored and a summary of the contributions presented.

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Section 6.1. Specification of a Simulation-Based CreditRisk+ Model

While presented in an alternative structure to that found in the asset value models reviewed in Chapter 4, CreditRisk+ can be shown to share common conceptual and mathematical foundations with those studied in Chapter 4. In Gordy (2000) a single sector implementation is used to present a comparative mapping between a simulation- based CreditRisk+ model and the AVM presented in Chapter 4. In this Section we elaborate on this comparison, while in Section 6.2 results based on various implementations of the simulation-based CreditRisk+ are presented.

The simulation-based AVM model of Chapter 4 presented borrower defaults as realizations of a normally distributed latent variable below some given default barrier. The latent variable was assumed to depend on a single systematic factor and an idiosyncratic shock, both generated from standard normal distributions. Correlations between obligors were determined through common dependence on the single systematic factor, and were calibrated from PD averages and volatilities observed in the Financing Company SME portfolio.

In contrast, the CreditRisk+ model does not place assumptions on the cause of default; in a single sector implementation, borrower default probabilities are modelled to vary over time, increasing or decreasing with a gamma-distributed latent systematic factor. Borrower probability of default co-movements with the systematic factor thereby

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generate correlations in defaults, Gordy (2000, p. 120). In this Section we will use methods presented in Gordy (2000) to present a simulation-baesd implementation of the CreditRisk+ framework. Using this framework we are able to form direct comparisons in the simulated portfolio loss distribution using two alternative calibrations to the single risk factor distribution and the sensitivities to that factor.

In the first calibration we follow Gordy (2000) and fix the risk factor paramters and calibrate the sensitivies accordingly, this method will be referred to as the unitary normalized sector volatility setting. In the second calibration we will hold sensitivities constant and calibrate the shape and scale parameters of the gamma-distributed latent factor from Financing Company default rate data; this setting will be referred to as the unitary weight setting. These calibrations, as in Chapter 4, will depend solely on the PD averages and volatilities observed in the Financing Company SME portfolio.

We construct the CreditRisk+ simulation-based implementation such that it uses the same simulation framework as that presented in Chapter 4. Analogously to Equation (5.11) and Equation (5.12), presented in Subsection 5.1.1, adhering to a single sector setting and dropping the “k” subscript of the sector parameters, we define a single gamma- distributed systematic factor (X) with shape and scale parameters:

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Allowing for the presence of an idiosyncratic factor with mean one and zero variance, for a given borrower (i) in risk grade (ζ) the probability of default, conditional on realizations of the systematic factor, is amplified or subdued according to a given sensitivity wi,ζ. More specifically, we write:

( ) ̅̅̅̅ ( ( )) ( )

where ̅̅̅̅ is the unconditional long term average probability of default for a given risk grade (ζ); see Gundlach (2004, pp. 16-17) and Gordy (2000, pp. 122-124); and:

∑ ̅̅̅̅ ∑ ̅̅̅̅̅̅̅̅̅ ( )

To generate a distribution of defaults, Gordy (2000, p. 127) suggests the following specification for the latent variable yi:

( ( ))

( )

such that the idioysyncratic risk factors are exponentially distributed with scale parameter equal to one. This specification is analogous to the AVM specification described in Subsection 4.1.1; see Equation (4.3). For a borrower (i), default occurs if and only if ̅̅̅̅ , where:

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( ̅̅̅̅ | ) ( ̅̅̅̅ ( ( ))| )

( ̅̅̅̅ ( ₍

)))

̅̅̅̅ ( ( )) ( ) ( )

Defaults can be simulated by drawing realizations of ( ) for each borrower and a realization of the systematic factor (x), and applying Equation (6.4). This method, while useful for descriptive purposes is known to generate a small approximation error and can be replaced by independent draws of a ( ( )) variable, using the last line

of Equation (6.5), above, and draws of systematic factor (x); see Gordy (2000, p. 142).

Losses are generated by multiplying the simulated borrower defaults by their exposures and their losses given default, individual losses are summed to obtain the portfolio loss for that draw of the systematic factor. The process is repeated times and the loss distribution is generated. Furthermore, we use the simulation-based allocation methodology presented Chapter 4 to present capital charges by segment. This process therefore describes a simulation-based implementation of the CreditRisk+ framework in which an alternative systematic risk factor engine is used within the same portoflio loss distribution framework as that applied in Chapter 4. We thereby limit differences in implementation to the systematic risk factor distributional assumptions and calibration

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techniques of the asset value model (AVM) presented in Chapter 4 and the CreditRisk+ model presented in Chapter 5 and extended in this Section.

As previously noted, under the AVM systematic and idiosyncratic risk factors were modeled as standard normal random variables, asset correlations representing obligor dependencies were then non-parametrically calibrated using historical PD and PD volatility values. In calibrating the simulation-based CreditRisk+ we return to a subject touched on in Chapter 5, that of the unitary weight setting versus the unitary normalized volatility setting. In particular, CreditRisk+ implementations allow for the choice of calibration of obligor sensitivities with given assumed distributional characteristics for the sector factor, or, alternatively, for the calibration of the factor scale parameters given fixed sensitivities. In this Subsection we will present a method incorporating both of these calibrations. Results based on these calibrations will be compared against each other as well as against those obtained under the AVM of Chapter 4. In addition, we will calibrate our model to reflect the boosted correlations obtained in Chapter 4. This exercise will be similar to Dietsch and Petey (2002).

In particular, we observe that for the AVM, the methods presented in Subsection 4.1.1 are used to calibrate asset correlations from default rate mean and volatility. For a single sector implementation of the CreditRisk+ framework, with a borrower-specific idiosyncratic factor, see Credit Suisse (1997, p. 52), we can define a given segment (ζ)’s variance around its probability of default as:

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[ ( )] [ ̅̅̅̅̅ (

( ))] ̅̅̅̅̅ [( )]

( ̅̅̅̅ ) ( ) ( ̅̅̅̅ ) ( )

such that for the normalized volatility (√ [ ( )] _⁄̅̅̅̅ _{) we can uniquely determine} the weights (wζ) or the normalized sector volatility ( ⁄ ); see Gordy (2000, p. 134):

√ [ ( )] ⁄̅̅̅̅ ( ) ( )

In Section 6.2 we present cases of the simulation-based CreditRisk+ implementation in which the normalized sector volatility is held fixed and the weights are calibrated, as well as cases in which the weights are fixed (at a value of one) and the sector volatility is calibrated. Recall, under the analytical implementations of CreditRisk+ the weights are always fixed at a value of one.

Boosted Asset Correlations

In order to provide comparability between boosted asset correlation results in Chapter 3 and those obtainable under the CreditRisk+ framework for the same boosted asset

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correlation values, we use the work in Gordy (2000) as a bridge, this is similar to the boosting and bridging in Dietsch and Petey (2002).

Specifically, in a single sector setting, for a given segment, we use Equation (4.10) to calibrate the default rate variance given the boosted asset correlation value and the segment PD. The next step then depends on our choice of calibration setting: Under the unitary normalized volatility setting we substitue this “boosted variance” into Equation (6.7) and thus obtain the segment weight; Under the unitary weight setting we take the square root of segment-specific boosted variances and substitute them into Equation (6.3) to obtain the systematic risk factor standard deviation. The boosted weight or sector volatility is then substituted into the simulation-based implementation of the CreditRisk+ framework described above to generate capital charges for the portfolio.

Final Comparative Model Specifications

In this exercise we use identical data to calibrate the CreditRisk+ and asset value models. For both models, data segmented by borrower and loan, LGD values are given as either 73% or 41%. In both models a conditional default framework is established such that a latent factor mimicking the business cycle is simulated, and borrowers default depending on a combination of their sensitivity to the systematic latent factor and idiosyncratic effects. Our database of historical SME defaults, spanning 12 years and containing information on borrower creditworthiness, size and industry, allows us to determine that sensitivity under the varying assumptions of the models implemented. Loss distributions

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are generated for various realizations of the latent and idiosyncratic factors for each obligor, a given condition for default, and fixed obligor-specific exposure and LGD values.

In order to combine the defaults with losses, we simply use the same simulation engine used Chapter 4, so that defaults are generated by borrower and then assigned to the corresponding loans with which LGD and exposure amounts are associated. Given the use of identical input data in the calibrations of the models presented in Chapter 3 and Chapter 4, we obtain results using the exact same inputs but with slight alternations to the conditional default and portfolio loss structure.

Section 6.2. Comparative Capital Charges: Basel II, AVM and CreditRisk+

Our results will review the comparability of various calibrations of the CreditRisk+ model to the AVM model in terms of correlations, loss distributions, and capital charges – both overall and at the segment-specific level. In turn, we present results along three Comparisons. Comparison I presents loss distributions under the AVM framework and various settings of the CreditRisk+ framework. Using Figure 6.1 and Table 6.1A we are able to identify settings of the CreditRisk+ framework under which calculated loss distributions are clearly divergent from that obtained under the AVM. Comparison II presents a comparison of segment-specific default correlations, along with other parameters, under the two frameworks and suggests a calibration refinement for

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CreditRisk+. Having observed comparative loss distribution results under the various settings and calibrations of the CreditRisk+ and AVM frameworks, we apply the boosting method described in Section 6.1 to compare EC results under both frameworks to those obtained under the Basel II AIRB framework for portfolio credit risk; see Case 2 of the Basel II implementations of Chapter 3. This comparison, which we will refer to as Comparison III, uses Figures 6.2 and 6.3, and Tables 6.4 to 6.6, to clearly delineate loss distributions and EC allocations under the boosted and non-boosted implementations of the frameworks.

Fixed normalized standard deviation ratio (σ/μ) settings of the Single Sector CreditRisk+ model include sector weights which have been calibrated according to Equation (6.7), such that for each segment, the sector weight and the idiosyncratic weight sum to one. The unitary sector weight setting of the Single Sector CreditRisk+ model includes a sector weight that has been set to one and the single sector ratio that has been calibrated from the data according to Equation (6.3). This specification, matches that used in the analytical implementation of the CreditRisk+ model and will be the primary base of comparison against the internally-calibrated AVM model. In addition, we will use this specification as a basis for comparisons using the boosted asset correlations of Chapter 4, using boosted asset correlations given in Table 4.7. This exercise shares similarities with Dietsch and Petey (2002).

Finally, we will examine default correlation results obtained in the CreditRisk+ framework and compare them to those obtained in the AVM and pre-calibrated into the

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Basel II. This work will be undertaken in Subsection 6.2.3. It should be noted that here we are interested in the default correlations among borrowers in the same portfolio segment and will not discuss the cross-correlations across segments. These values can be readily derived from results presented in this Subsection and the application of Equation (C.1) of Appendix C.

Figures presented in this Chapter will have two panels: the top panel will show loss distributions for “non-boosted” internally calibrated models; the lower panel will show loss distributions for “boosted” internally calibrated models. In Figure 6.2, a third (bottom) panel juxtaposes “boosted” and “non-boosted” model loss distributions, providing further perspective on results obtained under each calibration.

Comparison I Loss Distributions under various settings of CreditRisk+ Figure 6.1; Table 6.1A; Table 6.2

We examine loss distributions under various settings of the CreditRisk+ model, under an RG calibration, and compare them to a similarly calibrated AVM implementation. Settings include the unitary weight setting, and fixed normalized standard deviation settings with values for the ratio (σ/μ) of one, 0.5 and 0.25. Our choice is based on results our observation of normalized PD standard deviation values for our SME RG segments, as depicted in Table 4.4. As can been seen in Table 4.4, SME normalized PD standard deviation values are considerably lower to those observed for Corporate borrowers. These Corporate borrower normalized PD standard deviation values may

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suggest a calibration of one or greater for the CreditRisk+ single sector risk factor normalized standard deviation, a fact that has resulted in such settings throughout the literature – even when that literature has claimed to present settings suitable for SME