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**CHAPTER 2 ****A THEORETICAL FRAMEWORK FOR ESTIMATING A CONDITIONAL PORTFOLIO LOSS DISTRIBUTION: THE PESARAN, SCHUERMANN, TREUTLER AND WEINER (2006) METHODOLOGY**

**INTRODUCTION**

At the heart of the structural default models lies the Merton (1974) approach to default risk. As argued in chapter 1, a firm is expected to default if the underlying asset value (where assets value is equal to the sum of equity and debt) falls below a specific threshold level. The threshold level is determined by the underlying callable liabilities and will essentially change over time with changes in liability, equity and asset value. The fundamental insight of Merton (1974) is that debt can be regarded as a put option on the value of the assets of the firm i.e. debt holders sell a put option on the underlying firm’s assets to equity holders. If the value of the firm falls below this threshold level or strike price, the equity holders would effectively put the company to the debt holders. Typically, default is defined by banks and rating agencies as a nonpayment of any coupon or interest payment or any principal due, although stricter definitions such as delisting activities are sometimes also applied for internal use by banks.

As such, Merton (1974) was able to apply debt valuation to the option-valuation field for which increasingly sophisticated valuation techniques were becoming available.The option-value view of debt has since been applied in both credit and interest rate risk management techniques and is a fundamental part of financial risk management and portfolio management techniques and models (see for example Van Deventer,Imai and Mesler (2004)).Similar to the option valuation field, quantification of default risk will therefore require modelling of three aspects:

The evolution of the underlying firm’s value;

The default threshold of the specific firm; and

The degree to which firm value is correlated with other companies; and the macroeconomic environment, also known as the asset-value correlation.

Although equity value data is easily available for listed companies, the underlying asset value of companies is not readily available. In fact, most Merton-type models such as the Moody’s KMV Portfolio Manager™ uses a propriety database in which firm asset value is estimated from balance sheet and equity data. From this database,asset value correlation and volatility are estimated while the default thresholds are taken to be some function of the short-term and long-term debt in each period. In this framework, a distance to default variable for each company is estimated as the number of standard deviations the underlying firm value is, from the default threshold. Default probability is then inferred through historical observations of the distance to default and actual default experiences.Clearly, such a framework is heavily dependent on the propriety data underlying both the asset value evolution as well as the distance to default and default probability estimation. The methodology proposed by PSTW (2006) has made a significant advance in credit risk modelling in that it avoids the usage of proprietary balance sheet and distance to default data, instead focussing on credit ratings which are more freely available. By linking this adjusted structural default model to a structural global econometric (GVAR) model, credit risk analysis and portfolio management can be done through the use of a conditional loss distribution estimation and simulation process.

This chapter aims to provide a comprehensive discussion of the PSTW (2006) approach and argues that the methodology provides a foundation through which credit risk managers in South African financial markets can evaluate and analyse their portfolios not only from a domestic macroeconomic perspective, but by linking into the existing GVAR model will also be able to use a global perspective in portfolio analysis.

**AN ADJUSTED MERTON-BASED MODEL OF DEFAULT**

Consider a firm j in a country i having a total asset value of Vji,t at time t while the underlying debt obligation is designated by Dji,t . Using the Merton (1974) approach default would occur at maturity date of debt, t+H if the firm’s assets are less than the value of the debt, i.e. default occurs if Vji,t+H < Dji,t+H . Clearly, this default definition is similar to a European put option which is only exercisable at maturity. The first passage model proposed by Black and Cox (1976) allows default to occur the first time Vji,t < Dji,t over the time period t to maturity date t+H. The default probability is therefore determined by the probability distribution of asset values at the terminal date, t+H or over the period t to t+H in the Merton and first passage models respectively. Although the method proposed by PSTW (2006) can be adapted to suit both approaches it is applied here to the Merton European put option specification. Using the accounting definition equation, the value of a firm should equal the value of debt and equity, i.e.From equation 3 it is clear that default will only occur if the equity value of a firm is negative. This condition is rather restrictive and not necessarily true in practice. Often shareholders work pro-actively and put the firm up for receivership before the equity value of the firm hits zero. Several studies have also shown that equity owners receive some compensation even though debt holders have not been paid in full (Eberhart and Weiss (1998) and Longhofer (1997)) and data suggests that equity values stay positive even for insolvent firms (Betker (1995), Franks and Torous (1991) and LoPucki (1991)). From a bank perspective, various loan conditions allow banks to force firms into default if equity values fall below a specific non-negative threshold (Garbade (2001)). As argued by PSTW (2006), the value of equity does not only take into account firm asset value but also includes an option that firms may recover before creditors take control of assets. On the other hand, borrowers often work out refinancing arrangements if, for example, one or two coupon payments have been missed, avoiding the firm going into bankruptcy. As such PSTW (2006) assumes that default takes place if In equation 4, Cji,t+H represents some positive default threshold which can vary over time and differ between firms depending on firm-specific characteristics such as sector or industry classification, leverage, profitability, firm size or age and qualitative factors such as management style. Clearly, accounting-based factors such as leverage are measurable and obtainable through data vendors. However, although new accounting standards are increasingly trying to improve the quality of data provided through financial statement disclosures, such data is still noisy and includes information asymmetries between firm management and investors (WittenbergMoerman (2006)).In order to estimate economic capital through an economic capital model, one implicitly encounters the latter-mentioned measurement and information asymmetry problems as with a default model. PSTW (2006) attempts to alleviate these challenges by using credit ratings for firms (R). R may take the values usually depicted by either Moody’s (Aaa, Ba, Baa,…, Caa) or S&P’s (AAA, AA, BBB,…,CCC) rating notations. The use of ratings facilitates the estimation of default thresholds in order to obtain the default probabilities of each firm. Most rating agencies perform a rigorous process of interviews with firm officials and in-depth analysis of financial statements and observable market data to assign a particular rating to a firm. Moreover, rating agencies are explicit in their commitment to assigning consistent ratings between firms and also over time so that comparisons can be performed over longer periods.As such it is reasonable to assume that the information contained in the ratings outcome, R, contains estimates of current balance sheet and equity return data, historic return data and non-publicly available information. This is an attempt to bridge the information asymmetry gap and provide information on all the firms in the past which have been given a similar rating. Equation 7 illustrates that over the horizon H, the relative decline in firm value must be big enough to result in default. As such the default becomes independent of firm size in the default process, but is an important determinant of the initial rating assigned by the rating agencies to a specific firm. Naturally a small firm will need to hold a larger equity cushion relative to a big firm if it is likely to withstand a given shock.

As usual, if firm equity values follow equation 5, ln(ER,t+H/ERt) can be approximated by cumulative returns so that (7) becomes However, these different criteria will be dependent on data availability as well as the fact that a reasonable sample of data will be required within each sub-group to allow reliable estimations to be obtained.

**FIRM-SPECIFIC DEFAULTS**

Using this adapted Merton model of default it is now possible to specify firm-specific default conditions and a default probability model. Following the standard multifactor model specification, assume that the return of firm j in region i over the time period t to t+1, i.e. rji,t+1 can be decomposed into two portions, conditional on information available at time t, given the information set Ωt , i.e with ,tji the conditional mean and tji 1, the innovation component of the return process. Since the conditional mean of the return process can be forecasted using macroeconomic factors (or risk drivers) this creates the channel through which one can simulate the impact that economic shocks will have on firm returns. As usual the individual innovation components will be assumed to take on the normal Merton model form The assumption that the conditional variance of returns is time invariant has been heavily disputed throughout the literature in the case of high frequency data. As such techniques such as GARCH modelling which allows for time-dependent variances should be employed when dealing with such cases. Although the use of such techniques will slightly alter the current framework the general form is still preserved.In this application quarterly data will be employed and time-invariant return variances will be assumed to hold true.Although the default threshold ratios of R-rated firms are assumed to be the same, the default probabilities will differ conditional on the firm-specific return characteristics.

More formally, by analysing equation 17 firm j‘s default probability (and also its distance to default) will be driven by:

The firm’s credit rating at the beginning of the forecast period. Lower credit ratings will by “closer” to the default thresholds;

The volatility of equity returns, ji . The more volatile the more likely a firm is to cross the threshold at any time; and

The unconditional equity return, ,tji . The higher the expected return, the “further” the firm is from default.

A macroeconometric risk driver model (the GVAR model in the case of PSTW (2006)) provides a conditional loss distribution through the ability to link the changes in macroeconomic variables (in region i and globally) to firm-specific credit risk. The linkages in the macroeconomy are established through the interaction of the changes in macro-variables and firm-specific (conditional) means uji,t .

**CONDITIONAL CREDIT RISK MODELLING**

**THE MACROECONOMIC RISK DRIVER MODEL**

**Introduction**

The macroeconometric risk driver model (GVAR) used in PSTW (2006) comprises a total of 25 countries which are grouped into 11 regions and account for 80 per cent of world production. However, as stated by PSTW (2006) a cointegration framework can become computationally demanding and therefore seven key economies are modelled,namely the U.S., U.K., Japan, China, Germany, France and Italy alone while all other countries are modelled as part of regional groups, i.e. Western Europe, South East Asia, Latin America and the Middle East. Clearly, in the case of South Africa the GVAR model lacks applicability, since it does not include an African region.However, the approach is general enough so that country-specific cointegration models can be linked into the global and already established GVAR model. Therefore,the use of cointegration is applied in such a fashion that heterogeneity that exists across regions and countries is acknowledged.Although explained in detail below the basic intuition of the GVAR model is that specific vector error-correcting models (VECM), which relate macroeconomic variables such as gross domestic product GDP, inflation, interest rates, money supply,exchange rates and equity prices to foreign variables through the linkages of international trade patterns, are estimated for each region or country. Within the GVAR framework there are three channels through which economies and factors in each economy interact:

Contemporaneous dependence of domestic and foreign variables;

Dependence of country-specific variables on common global effects such as oil prices; and

Weak cross-sectional dependence of idiosyncratic shocks.

In this study a country-specific macroeconometric risk driver engine which can feed into the GVAR model and framework proposed by PSW (2004) will be constructed.

This will allow conditional loss estimation of a South African-specific credit portfolio but also opens the door for credit portfolio modelling on a global scale,because such a model can easily be linked into the GVAR model. In order to estimate and provide such a South African specific model it is necessary to analyse the construction of the GVAR model proposed by PSW (2004).

**LIST OF FIGURES **

**LIST OF TABLES**

**LIST OF ABBREVIATIONS **

**1 CREDIT RISK AND PORTFOLIO MODELLING **

**1.1 INTRODUCTION **

**1.2 DEFINING CREDIT RISK AND PORTFOLIO MODELLING **

**1.3 STRUCTURAL DEFAULT MODELS **

**1.4 POSITIONING THE STUDY WITHIN THE LITERATURE **

**1.5 METHODOLOGY **

**1.6 DATA **

**1.7 OUTLINE OF THE STUDY**

**2 A THEORETICAL FRAMEWORK FOR ESTIMATING A ****CONDITIONAL PORTFOLIO LOSS DISTRIBUTION: THE ****PESARAN, SCHUERMANN, TREUTLER AND WEINER (2006) ****METHODOLOGY**

**2.1 INTRODUCTION **

**2.2 AN ADJUSTED MERTON-BASED MODEL OF DEFAULT **

**2.3 FIRM-SPECIFIC DEFAULTS **

**2.4 CONDITIONAL CREDIT RISK MODELLING**

2.4.1 The macroeconomic risk driver model

2.4.1.1 Introduction

2.4.1.2 The global error-correcting macroeconometric model

2.4.1.3 Individual-country VECMs

2.4.1.4 Solving the GVAR model

2.4.1.5 Short-run dynamics of the global model

2.4.1.6 Forecasting and dynamic properties of the GVAR model

2.4.1.7 Impulse response and scenario analysis

2.4.2 Using individual-country models to construct the GVAR model

**2.5 FIRM-SPECIFIC RETURN DYNAMICS **

**2.6 LOSS GIVEN DEFAULT AND EXPECTED LOSS ****ESTIMATION **

**2.7 SIMULATING THE LOSS DISTRIBUTIONS**

**2.8 CONDITIONAL DEFAULT AND CONDITIONAL EXPECTED ****LOSS **

**2.9 MULTI-PERIOD LOSS DISTRIBUTIONS **

2.9.1 Baseline multi-period loss distribution

2.9.2 Conditional multi-period loss distribution

**2.10 CONCLUSION**

**3 AN EMPIRICAL APPLICATION OF A SOUTH AFRICAN ****GLOBAL ERROR CORRECTING MACROECONOMETRIC ****MODEL**

**3.1 INTRODUCTION **

**3.2 DATA SOURCES AND GLOBAL VARIABLE TIME SERIES ****CONSTRUCTION **

3.2.1 The trade weights

3.2.2 Global macroeconomic variables and data sources

3.2.3 Domestic and exogenous macroeconomic variables and data sources

**3.3 INTEGRATION PROPERTIES OF THE TIME SERIES **

**3.4 COINTEGRATING RANK PROPERTIES OF THE SYSTEM **

**3.5 IMPOSING IDENTIFICATION RESTRICTIONS ON THE ****VECM SYSTEM **

**3.6 ESTIMATION RESULTS **

**3.7 RESIDUAL DIAGNOSTICS AND VECM STABILITY TEST **

**3.8 DYNAMIC PROPERTIES OF THE VECM **

**3.9 STOCHASTIC FORECAST PROPERTIES OF THE VECM **

**3.10 CONCLUSION **

**4 DEFAULT RISK AND CONDITIONAL CREDIT PORTFOLIO ****LOSS SIMULATION **

**4.1 INTRODUCTION **

**4.2 DEFAULT THRESHOLDS BY RATING CATEGORY **

**4.3 THE SAMPLE PORTFOLIO **

**4.4 FIRM-SPECIFIC RETURN REGRESSIONS**

4.4.1 Pooled panel data model for portfolio-wide return estimates

4.4.2 Fixed and random effects panel data model for portfolio-wide return estimates

4.4.3 Firm-specific multi-factor models

**4.5 CONDITIONAL LOSS ESTIMATION AND SCENARIOANALYSIS RESULTS**

**4.6 CONCLUSION **

**5 SUMMARY AND CONCLUDING REMARKS**

**5.1 INTRODUCTION **

**5.2 SUMMARY RESULTS OVERVIEW **

5.2.1 Basic model outline and methodology

5.2.2 Empirical results

**5.3 AREA OF FUTURE RESEARCH **

**5.4 CONCLUSION **

**BIBLIOGRAPHY **

**APPENDIX A: FIRM-SPECIFIC RETURN MODELS**

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A MACROECONOMETRIC FRAMEWORK FOR CREDIT PORTFOLIO MODELLING IN SOUTH AFRICA