Credit Risk. KMV-Approach

Contents

List of Tables

List of Figures

List of Abbreviation

1 The Need for Credit-Risk Measurement
1.1 Basel Accords
1.2 Risk in the Three Pillars

2 Credit Risk
2.1 Major Risk Components
2.2 Credit Risk Models

3 KMV
3.1 Merton Approach
3.2 Characteristics KMV Model
3.3 KMV Approach
3.3.1 Value of Assets
3.3.2 Default Point
3.3.3 Distance to Default
3.3.4 Expected Default Frequency (EDF)
3.3.5 Loss Distribution
3.3.6 Default Predictive Power

4 Risk Measurement in Practice

Attachment

References

List of Tables

Table 1: “Ideal conditions“ in the market (Black & Scholes)

Table 2: Characteristics Merton Model (Kealhofer)

Table 3: Characteristics KMV Model

List of Figures

Figure 1: Three Pillars

Figure 2: Today’s Best-Practice Industry Models

Figure 3: Black & Scholes (1973)

Figure 4: Distance to default (I)

Figure 5: Distance to default (II)

Figure 6: EDF - DD

Figure 7: Comparison of survey results

Figure 8: Survey 2002

List of Abbreviations

Abbildung in dieser Leseprobe nicht enthalten

1 The Need for Credit-Risk Measurement

“ Until the 1990s, corporate credit analysis was viewed as an art rather than a science because analysts lacked a way to adequately quantify absolute levels of default risk. In the past decade, however, a revolution in credit-risk measurement has taken place. […]“

[Stephen Kealhofer, 2003]¹

1.1 Basel Accords

On January 1st 2007 the European directives (2006/48/EG)² and (2006/49/EG)³ con- cerning minimum capital requirements, which are equivalent to the new Capital Ac- cords elaborated from the Basel Committee on Banking Supervision (BCBS), were put into German national legislation.⁴ The basic aim of the Basel Accords is to gear banks’ capital requirements more closely than in the past to the actual economic risk, which should improve security and soundness of the financial system.⁵ Thus an accord be- comes operative, which had its beginning in the year 1988 with the publication of the Basel Accord (Basel I). The BCBS is retaining key elements of the capital adequacy framework (1988), but the revised framework gives greater consideration of assess- ments of risk provided by banks’ internal systems as input to capital calculations.⁶

1.2 Risk in the Three Pillars

The International Convergence of Capital Measurement and Capital Standards, in the term paper now named as the well-known Basel II, provides three pillars each dealing with risk, which can be seen in figure 1.⁷

As you can see in figure 1 differ- ent approaches are provided to deal with credit risk, either the Standardized Approach or the In- ternal Ratings Based Approach (IRB), which includes again two broad approaches, namely the foundation and the advanced ap- proach. One reason for banks to prefer IRB in opposition to the Standardized Approach is that a more specific approach tends to result in a lower capital backing with the same degree of risk.⁸

2 Credit Risk

A major frontier in modern finance is the quantification of credit risk.⁹ To get a good understanding of quantification of credit risk it would be advantageous to have a good understanding of these central inputs to credit portfolio models. After a short introduction in important risk parameters I would like to present the risk measurement approach “KMV - Model”, a version of the “Merton Model”.¹⁰

2.1 Major Risk Components

The three key Basel II risk parameters are probability of default (PD), loss given default (LGD) and exposure at default (EAD).¹¹

A loan commitment is marked as a default when it meets either or both of the two con- ditions:¹²

-The bank considers that the obligor is unlikely to pay its credit obligations to the banking group in full, without recourse by the bank to action such as realiz- ing security (if held).
-The obligor is past due more than 90 days on any material credit obligation to the banking group.

The probability of default measures the probability of a loan default in a given period. PD’s are not predictable and so they are more precisely estimations of PD’s in the statistical sense.¹³ For assigning PD’s to customers in a credit portfolio two approaches can essentially be taken, either a calibration of PD’s to ratings, which results in a mapping or a calibration from market data, where the Expected Default Frequencies (EDF) determined from the KMV model is the most famous representative.¹⁴

Exposure at Default is the exposure the bank has to its borrower. The EAD can be seen as the sum of the outstanding and the Credit Conversion Factor (CCF), which is e.g. 75% in the foundation approach¹⁵ and stand for the expected portion of the commitments likely to be drawn prior to [Abbildung in dieser Leseprobe nicht enthalten]¹⁶

If a default happens the LGD is the proportion of the EAD the lender really loses. The connection between LGD and recovery rate can be shown as

Abbildung in dieser Leseprobe nicht enthalten^{17 18}

and refers further to the expectation of the severity:[Abbildung in dieser Leseprobe nicht enthalten]¹⁹

A model that takes the correlation between recoveries and default risk into account is Moody's KMV LossCalc™²⁰

Due to the three key risk parameters and the acceptance of simplifying assumption a formula for the expected loss (EL) can be written in its most simple representation:

Abbildung in dieser Leseprobe nicht enthalten²¹

2.2 Credit Risk Models

To get a better understanding about different models for measuring credit portfolio risk and to classify the KMV model, which will be discussed later in this term paper, figure 2 shows a quick overview about the classification of best-practice industry models.

Abbildung in dieser Leseprobe nicht enthalten

Figure 2: Today’s Best-Practice Industry Models²²

3 KMV

A de facto standard for default-risk measurement in the world of credit risk is a novel approach of the Merton model taken in 1984 and extended by the KMV Corporation,²³ which was founded in 1990 and now belonging to the rating agency Moody’s.²⁴ To get a basic idea of the KMV model a short presentation of the Merton approach might be helpful, because the tools of KMV Corporation, namely (Credit MonitorTM) for estimating default probabilities and (Portfolio ManagerTM) for managing credit portfolios are based on a modification of Merton’s asset value model.²⁵

The Asset Value Model (AVM), an important contribution to modern finance goes back in its original to Merton and Black & Scholes.²⁶ As you can see in figure 2 the KMV is an Asset Value Model.

3.1 Merton Approach

Black & Scholes, Nobel laureates²⁷ for a new method to determine the value of deriva- tives in 1997, derived a formula for the value of an option in terms of the price of the stock. The ideal conditions in the market assumed by Black & Scholes are shown in table 1.²⁸

Abbildung in dieser Leseprobe nicht enthalten²⁹

Further information about denotation and the cumulative normal density function are given in figure 3. Due to this seminal approach one could view the equity of a company as a call option.³⁰

The Black & Scholes formula can be seen as coherent framework for the objective measurement of credit risk. But this formula would only be appropriate for a firm with liabilities consisting only of a single class of debt, which pay no interest prior to maturity and furthermore no dividend paying equity.³¹

Due to further elaborations by Merton (1973, 1974), Black and Cox (1976) an Ingersoll (1977a) the emerging approach is known as “Merton model “.³²

Thus it is possible to price almost any type of financial instrument with inputs, which are on the whole observable.³³ Characteristics of this model presented by Kealhofer can be seen in table 2.³⁴ From original insights of Black-Scholes-Merton, Kealhofer and Vasicek developed a straightforward variant of the abovementioned Merton model.³⁵

3.2 Characteristics KMV Model

According to Kealhofer the KMV Model has the following characteristics³⁶, which can be seen in table 3. As above mentioned that KMV model is largely a variant of Merton Model it differs from this canonical Merton model in some significant ways. If you compare the characteristic number 1 in table 2 and in table 3, you will see that the KMV model allows any number of debt and nondebt fixed liabilities, whereas in the Merton model is only allowed a single debt liability. Thus this aspect offers the KMV model more possibilities. Also characteristic number 4 (table 3) compared to number 3 (table 2) opens up more possibilities in the KMV-Model, because cash payouts can be made. Compared to CreditMetrics the KMV approach doesn’t rely on the fact that firms in the same rating class have the same default rate.³⁷ Due to that it is possible to get a more precise method to measure risk.

3.3 KMV Approach

In contrast to CreditMetrics, another asset value model, the KMV framework comprises following assumptions for the specificity of each issuer:³⁸

-own default probability
-own asset return distribution
-own capital structure

So the KMV framework is based on Expected Default Frequency (EDF) for each posi- tion in the credit portfolio.³⁹ The EDF is a crucial indicator of default, because they tend to shoot up quickly until default occurs,⁴⁰ EDF, the actual probability of default, is a measure of the probability that a firm will default over a specified period of time⁴¹ and is a function of the capital structure, the volatility of the asset returns and the current asset value.⁴²

According to Moody’s a firm’s EDF credit measure is determined by three key values:⁴³

-The current market value of the firm (market value of assets) (point 3.3.1)
-The level of the firm’s obligation (default point) (point 3.3.2)
-The vulnerability of the market value to large changes (asset volatility), which is derived with an iterative technique of KMV.⁴⁴

As seen in table 3 (Characteristics - KMV model) the capital structure comprises equity (Nr. 1, table 3), short and long term-debt (Nr. 3, table 3), which all can make fixed cash payouts (Nr. 4, table 3) and the structure may also contain other capital instruments (Nr. 2, table 3).

3.3.1 Value of Assets

The asset value of a firm can be derived either, if all liabilities were traded and marketto-market every day, through the sum of the market values of the firm’s liabilities⁴⁵ or through the option pricing model.⁴⁶

Due to the simplifying assumptions in table 3 it is possible to write the value of equity VE as follows⁴⁷

[Abbildung in dieser Leseprobe nicht enthalten]⁴⁸

and thus the asset value VA as

[Abbildung in dieser Leseprobe nicht enthalten]⁴⁹

If the observed data of equity price and its volatility are given the asset value can be determined with the aid of the balance sheet according to the option pricing theory.⁵⁰ The asset value is the first key value of EDF according abovementioned to Moody’s.

3.3.2 Default Point

In the default-only mode of thee KMV model a firm is considered to be in default if the asset value at a certain time horizon is below a crucial threshold.

Abbildung in dieser Leseprobe nicht enthalten⁵¹

Thus in the KMV EDF model a firm defaults, when the value of the ongoing business (market value) falls below its liabilities payable (default point),⁵² which is a point set at par value of current liabilities including short term debt plus half the long term debt.

3.3.3 Distance to Default

A formula, which computes an index “distance-to-default” (DD), an intermediate phase before computing the probabilities of default, which is the number of standard devia- tions between the mean of the distribution of the asset value and default point (DPT) is shown in figure 4.⁵³

An example of a less risky (Firm I) and a more risky company (Firm II) should exemplify the resulting differences in DD, according to the formula in figure 3.⁵⁴

Abbildung in dieser Leseprobe nicht enthalten

The DD depends on the market value of assets, the DPT and the asset volatility. Thus the DD depends on the same three variables as the firm’s EDF credit measure (Moody’s) mentioned under point 3.3 (KMV Approach). The higher the value of long- term debt, short term debt, volatility of asset return and the lower the expected asset value the lower is the distance-to-default. The lower the value of the DD the higher is the possibility of a default of the specific company. Furthermore it is possible to define DD, expressed in unit of asset return standard deviation at time horizon T, as the num- ber of standard deviations between the mean of the distribution of the asset value and the DPT. Another formula is shown in figure 5.⁵⁵ In this formula the DD is expressed in unit of asset return standard deviation at time horizon T. Thus after computing the dis- tance-to-default, implying the market value of assets and their volatility, it can be mapped to actual probabilities of default, for a given time horizon.⁵⁶

3.3.4 Expected Default Frequency (EDF)

Based on the fact that the asset market value and its volatility are known the probability that the asset value will decline to the default point. This probability is named as the EDF credit meas- ure.⁵⁷ The connection between EDF and DD can be seen in figure 6 presented by Crouhy, M. et al.⁵⁸ In this figure you can see, that the EDF is increasing, if the distance-to-default is getting smaller.⁵⁹

Figure 6: EDF - DD

Abbildung in dieser Leseprobe nicht enthalten

The nearer the firm goes to the default point the higher is the expected default fre- quency. In the example the population of all firms with a DD of 4 at one point in time are considered. If e.g. 40 firms default in one year, an estimation based on historical data, and the considered population is 10.000 firms then the EDF can be written as:

Abbildung in dieser Leseprobe nicht enthalten

After computing the EDF one can compare ratings presented e.g. by Moody’s or by Standard & Poors.⁶⁰ In a survey in 2002 financial institu- tions were asked to compare the probabilities of default, reported by the respondents, with the EDFTM obtained from KMV’s CreditMoni- torTM. According to this figure one can see that the Median Moody’s- KMV EDF^TM is higher as the survey results in the good rating classes.

Abbildung in dieser Leseprobe nicht enthalten

Figure 7: Comparison of Survey Results with Other Probability of Default Measures

Within the class BB/Ba the EDFTM is nearly the same size as the median in the survey.

3.3.5 Loss Distribution

The loss distribution of a credit portfolio is the main output of (Portfolio ManagerTM).⁶¹ Based on the fact, that the exposure amount is given the expected credit losses of an instrument can be estimated more precisely in combining results for LGDs with EDFs than investigating EDFs on its own.⁶² Because of the better possibilities for estimation Moody's KMV LossCalc™ supports institutions to manage their risks in a more sophis- ticated way and to give a support in meeting the Basel II requirements, if an Internal Rating Based Approach is used.⁶³

3.3.6 Default Predictive Power

For each statistical model it is necessary to have tests, which can predict the power of the model. The “power tests” characterize the relative ability of a default-risk measure to correctly identify companies that subsequently default versus incorrectly identified companies. Another sort of analysis, namely “Intracohort analysis” provides a method for evaluating whether differences between two measures are result of additional infor- mation or simply noise.⁶⁴

4 Risk Measurement in Practice

In a survey of credit portfolio management practices the question was asked what mod- els are used as primary credit capital portfolio model. It appeared that with 40 % the financial institutions marked Moody’s-KMV Portfolio Manager as their primary credit capital/portfolio model.⁶⁵ But the internally-developed models (other than a macro fac- tor model) were chosen to 43%. All in all one can summarize that according to the sur- vey in 2004 Moody’s-KMV Portfolio Manager was the most used model, which is ex- ternal developed, but the most used models are the internally-developed models.

In a previous survey in 2002⁶⁶ it can be seen that Moody’s-KMV Portfolio Man- ager is with 69% chosen as the primary model used from financial institutions. Internally-developed models (other than a macro factor model) are only to 17% in- dicated as primary model.

Putting these results together one can say Figure 8: Survey 2002 that internally-developed models make inroads and that external developed models seem to be used in fewer cases.

The credit analyses, which are named from Kealhofer as an art (compare section 1) until the 1990s the revolution in credit risk measurement has taken place. Due to the devel- opment in portfolio management you can derive at least on safe prediction. The way that loans and other credits will be managed in the future will be very different from the way it was done in the past.⁶⁷

Attachment

Table 1⁶⁸

“Ideal conditions“ in the market (Black & Scholes)

1. The short term interest rate is known and is constant through time.
2. The stock price follows a random walk in continuous time with a variance rate proportional to the square of the stock price. Thus the distribution of possible stock prices at the end of any finite interval is lognormal. The variance rate of the return on the stock market is constant.
3. The stock pays no dividends or other distributions.
4. The option is „European,“ that is, it can only be exercised at maturity.
5. There are no transaction costs in buying or selling the stock or the option.
6. It is possible to borrow any fraction of the price of a security to buy it or to hold it, at the short term interest rate.
7. There are no penalties to short selling. A seller who does not own a secu- rity will simply accept the price oft he security from a buyer. And will agree to settle with the buyer on some future date by paying him an amount equal to the price of the security on that date.

Table 2⁶⁹

Characteristics - Merton Model (Kealhofer)

1. The company has a single debt liability, has equity and has no other obligations.
2. The liability promises a continuous fixed coupon flow and has an infinite maturity.
3. The company makes no other cash payouts (e.g. equity dividends).

Table 3⁷⁰

Characteristics - KMV Model

1. The company may have, in addition to common equity and possible preferred stock, any number of debt and nondebt fixed liabilities.
2. The company may have warrants, convertible debt, and/or convertible preferred stock.
3. Obligations may be short term, in which case they are treated as demandable by creditors, or long term, in which case they are treated as perpetuities.
4. Any and all classes of liability, including equity, may make fixed cash payouts.
5. If the market value of the company’s assets falls below a certain (the de- fault point), the company will default on its obligations; this default point depends on the nature and extent of the company’s fixed obligations.
6. Default is a company-wide event, not an obligation-specific event.

Figure 3⁷¹

Abbildung in dieser Leseprobe nicht enthalten

Denotation:

Abbildung in dieser Leseprobe nicht enthalten

Figure 4⁷²

Distance-to-default (I)

Abbildung in dieser Leseprobe nicht enthalten

Figure 5⁷³

Distance-to-default (II)

Abbildung in dieser Leseprobe nicht enthalten

References

Bank For International Settlements. June 2004. International Convergence of Capital Measurement and Capital Standards - A Revised Framework. [Internet]. Available at: http://www.bis.org/publ/bcbs107.pdf?noframes=1 [accessed 04 November 2007].

Black, F., and Scholes, M., 1973. The Pricing of Options and Corporate Liabilities. Journal of Political Economy, [Online]. 81 (3), p. 637 - 659.

Bluhm, C., Overbeck, L. & Wagner, C., 2003.

An Introduction to CREDIT RISK MODELING, Chapman & Hall/CRC.

Crouhy, M., Galai, D. & Mark, R., 2000. A comparative analysis of current credit risk models. Journal of Banking & Finance, [Online]. 24 (1/2), p. 59 - 117.

Deutsche Bundesbank. Basel II - the new Capital Accord, [Internet]. Available at: http://www.bundesbank.de/bankenaufsicht/ bankenaufsicht_basel.en.php [accessed 04 November 2007].

Deutsche Bundesbank. Solvency, [Internet]. Available at: http://www.bundesbank.de/bankenaufsicht/ bankenaufsicht_eigen.en.php [accessed 04 November 2007].

Deutsche Bundesbank. Neue Transparenzregeln für Kreditinstitute, [Internet]. Available at: http://www.bundesbank.de/download/volkswirtschaft/mba/ 2005/200510mba_transparenzregeln.pdf [accessed 07 November 2007]

Directive 2006/48/EC of the European Parliament and of the Council of 14 June 2006 relating to the taking up and pursuit of the business of credit institutions (recast). Available at: http://eur-lex.europa.eu/LexUriServ/site/en/oj/2006/l_177/ l_17720060630en00010200.pdf [accessed 04 November 2007].

Directive 2006/49/EC of the European Parliament and of the Council of 14 June 2006 on the capital adequacy of investment firms and credit institutions (recast). Available at: http://eur-lex.europa.eu/LexUriServ/site/en/oj/2006/l_177/ l_17720060630en02010255.pdf [accessed 04 November 2007].

Engelmann, B., and Rauhmeier, R., 2006. The Basel II Risk Parameters - Estimation, Validation, and Stress Testing. Springer Berlin Heidelberg.

Henking, A., Bluhm, C. & Fahrmeir, L., 2006. Kreditrisikomessung. Springer Berlin Heidelberg.

Kealhofer, S., 2003. Quantifying Credit Risk I: Default Prediction. Financial Analysts Journal, [Online]. 59 (1), p. 30-44.

Kealhofer, S., 2003. Quantifying Credit Risk II: Debt Valuation. Financial Analysts Journal, [Online]. 59 (3), p. 78-92.

Merton, R., 1974. On the pricing of corporate debt: The risk structure of interest rates. The journal of Finance, [Online]. 29 p. 449 - 470.

Moody’s[Abbildung in dieser Leseprobe nicht enthalten] EDF Overview, [Internet].

Available at: http://www.moodyskmv.com/newsevents/files/EDF_Overview.pdf [accessed 12 November 2007].

Moody’s [Abbildung in dieser Leseprobe nicht enthalten] LOSSCALC V2: DYNAMIC PREDICTION OF LGD,

[Internet]. Available at: http://www.moodyskmv.com/research/newResearch_wp.html [accessed 29 November 2007].

Moody’s [Abbildung in dieser Leseprobe nicht enthalten] Methodology Frequently Asked Questions, [Internet]. Available at: http://www.moodyskmv.com/research/faq.html# [accessed 29 November 2007].

Moody’s [Abbildung in dieser Leseprobe nicht enthalten]. STAND-ALONE CREDIT RISK MEASURES, [Internet]. Available at: http://www.moodyskmv.com/products/standAlone_ creditRisk.html [accessed 04 December 2007].

Nobelprice.org, [Internet]. Available at: http://nobelprize.org/nobel_prizes/economics/laureates/1997/ [accessed 08 November 2007].

Rutter Associates LLC. 2004 RUTTER ASSOCIATES SURVEY OF CREDIT PORTFOLIO MANAGEMENT PRACTICES, [Online]. Available at: http://www.rutterassociates.com/pdf/EXCERPT_Survey_of_ CPM_Practices_2004.pdf [accessed 04 December 2007].

Smithson, C., et al. Results from the 2002 Survey of Credit Portfolio Management Practices, [Online]. Available at: http://www.rutterassociates.com/pdf/Overview_of_2002_Survey.pdf [accessed 04 December 2007].

[...]

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¹ Cp. Kealhofer, S., Quantifying Credit Risk I: Default Prediction, p. 30.

² Directive 2006/48/EC.

³ Directive 2006/49/EC.

⁴ Cp. Deutsche Bundesbank, Solvency.

⁵ Cp. Deutsche Bundesbank, Basel II - the new Capital Accord.

⁶ Cp. Bank For International Settlement, International Convergence of Capital Measurement and Capital Standards, p. 2, para. 6.

⁷ Cp. Bank For International Settlement, International Convergence of Capital Measurement and Capital Standards, p. 6.

⁸ Cp. Henking, A., Kreditrisikomessung, p. 4.

⁹ Cp. Kealhofer, S., Quantifying Credit Risk II: Debt Valuation, p. 78.

¹⁰ Cp. Kealhofer, S., Quantifying Credit Risk I: Default Prediction, p. 30.

¹¹ Cp. Engelmann, B., The Basel II Risk Parameters, Preface.

¹² Bank For International Settlement, International Convergence of Capital Measurement and Capital Standards, p. 92 - 93, para. 452.

¹³ Cp. Henking, A., Kreditrisikomessung, p. 16 - 17.

¹⁴ Cp. Bluhm, C. et al, An Introduction to CREDIT RISK MODELING, p. 18.

¹⁵ Bank For International Settlement, International Convergence of Capital Measurement and Capital Standards, p. 92 - 93, para. 312.

¹⁶ Cp. Bluhm, C. et al, An Introduction to CREDIT RISK MODELING, p. 24.

¹⁷ Cp. Henking, A., Kreditrisikomessung, p. 20. Denotation: outstanding = OUT; commitments = COMM; Ȗ = CFF

¹⁸ Cp. Engelmann, B., The Basel II Risk Parameters, p. 129. Denotation: LGDt(i) and Rt(i) denote the LDG and recovery rate of bond i that defaults in year t, i =1,…, n t. The number of defaulted bonds in year t, t=1,…, T is denoted with n t.

¹⁹ Cp. Bluhm, C. et al, An Introduction to CREDIT RISK MODELING, p. 27.

²⁰ Cp. Moody’s[Abbildung in dieser Leseprobe nicht enthalten] STAND-ALONE CREDIT RISK MEASURES.

²¹ Cp. Bluhm, C. et al, An Introduction to CREDIT RISK MODELING, p. 17.

²² Cp. Bluhm, C. et al, An Introduction to CREDIT RISK MODELING, p. 67.

²³ Cp. Kealhofer, S., Quantifying Credit Risk I: Default Prediction, p. 30.

²⁴ Cp. Kealhofer, S., Quantifying Credit Risk I: Default Prediction, p. 30.

²⁵ Cp. Bluhm, C. et al, An Introduction to CREDIT RISK MODELING, p. 42.

²⁶ Cp. Bluhm, C. et al, An Introduction to CREDIT RISK MODELING, p. 123.

²⁷ Cp. Nobelprize.org

²⁸ Cp. Black, S., Scholes, M., The Pricing of Options and Corporate Liabilities, p. 640.

²⁹ Cp. Black, S., Scholes, M., The Pricing of Options and Corporate Liabilities, p. 644.

³⁰ Cp. Kealhofer, S., Quantifying Credit Risk I: Default Prediction, p. 30.

³¹ Cp. Moody’s[Abbildung in dieser Leseprobe nicht enthalten] Methodology Frequently Asked Questions.

³² Cp. Kealhofer, S., Quantifying Credit Risk I: Default Prediction, p. 30.

³³ Cp. Merton, R., On the pricing of corporate debt: The risk structure of interest rates, p. 452.

³⁴ Cp. Kealhofer, S., Quantifying Credit Risk I: Default Prediction, p. 31.

³⁵ Cp. Kealhofer, S., Quantifying Credit Risk I: Default Prediction, p. 42.

³⁶ Cp. Kealhofer, S., Quantifying Credit Risk I: Default Prediction, p. 32.

³⁷ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 84.

³⁸ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 65.

³⁹ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 85.

⁴⁰ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 92.

⁴¹ Cp. Moody’s[Abbildung in dieser Leseprobe nicht enthalten] EDF Overview.

⁴² Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 85.

⁴³ Cp. Moody’s[Abbildung in dieser Leseprobe nicht enthalten] EDF Overview.

⁴⁴ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 88.

⁴⁵ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 87.

⁴⁶ Cp. Merton, R., On the pricing of corporate debt: The risk structure of interest rates, p. 452.

⁴⁷ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 88.

⁴⁸ Denotation: VE = value of equity; ߪA volatility of asset return, K = leverage ratio in capital structure; c = average coupon paid on long-term debt; r = risk free interest rate.

⁴⁹ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 88.

⁵⁰ Cp. Moody’s[Abbildung in dieser Leseprobe nicht enthalten] Methodology Frequently Asked Questions.

⁵¹ Cp. Bluhm, C. et al, An Introduction to CREDIT RISK MODELING, p. 69. Denotation: company (݅); considered valuation horizon = T; Asset value = A; critical threshold C݅

⁵² Cp. Moody’s[Abbildung in dieser Leseprobe nicht enthalten] EDF Overview.

⁵³ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 88.

⁵⁴ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 92.

⁵⁵ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 89 - 90.

⁵⁶ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 90.

⁵⁷ Cp. Moody’s[Abbildung in dieser Leseprobe nicht enthalten] Methodology Frequently Asked Questions.

⁵⁸ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 91.

⁵⁹ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 92.

⁶⁰ Cp. Smithson, C., et al, Results from the 2002 Survey of Credit Portfolio Management Practices, p. 5.

⁶¹ Cp. Bluhm, C. et al, An Introduction to CREDIT RISK MODELING, p. 42.

⁶² Cp. Moody’s[Abbildung in dieser Leseprobe nicht enthalten] LOSSCALC V2: DYNAMIC PREDICTION OF LGD, p. 1.

⁶³ Cp. Moody’s[Abbildung in dieser Leseprobe nicht enthalten] STAND-ALONE CREDIT RISK MEASURES.

⁶⁴ Cp. Kealhofer, S., Quantifying Credit Risk I: Default Prediction, p. 33.

⁶⁵ Cp. Rutter Associates LLC, 2004 RUTTER ASSOCIATES SURVEY OF CREDIT PORTFOLIO MANAGEMENT PRACTICES, p. 5.

⁶⁶ Cp. Smithson, C., et al, Results from the 2002 Survey of Credit Portfolio Management Practices, p. 7.

⁶⁷ Cp. Rutter Associates LLC, 2004 RUTTER ASSOCIATES SURVEY OF CREDIT PORTFOLIO MANAGEMENT PRACTICES, p. 1.

⁶⁸ Cp. Black, S., Scholes, M., The Pricing of Options and Corporate Liabilities, p. 640.

⁶⁹ Cp. Kealhofer, S., Quantifying Credit Risk I: Default Prediction, p. 31.

⁷⁰ Cp. Kealhofer, S., Quantifying Credit Risk I: Default Prediction, p. 32.

⁷¹ Cp. Black, S., Scholes, M., The Pricing of Options and Corporate Liabilities, p. 644.

⁷² Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 89.

⁷³ Cp. Crouhy, M. et al, A comparative analysis of current credit risk models, p. 90.