FRIEDRICH-ALEXANDER UNIVERSITY ERLANGEN-NUREMBERG

CHAIR OF STATISTICS AND ECONOMETRICS

Term paper

Winter semester 2007/08

,,Credit Risk ­ KMV - Approach"

Schott, Robert

Semester: 8

Business Economics

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I

Contents

Contents I

List of Tables II

List of Figures III

List of Abbreviation IV

1

The Need for Credit-Risk Measurement 1

1.1 Basel Accords 1

1.2 Risk in the Three Pillars 2

2

Credit Risk 2

2.1 Major Risk Components 2

2.2 Credit Risk Models 4

3

KMV 4

3.1 Merton Approach 5

3.2 Characteristics KMV Model 6

3.3 KMV Approach 6

3.3.1 Value of Assets 7

3.3.2 Default Point 8

3.3.3 Distance to Default 8

3.3.4 Expected Default Frequency (EDF) 9

3.3.5 Loss Distribution 11

3.3.6 Default Predictive Power 11

4

Risk Measurement in Practice 11

Attachment 13

References 17

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II

List of Tables

Table 1: "Ideal conditions" in the market (Black & Scholes) 13

Table 2: Characteristics Merton Model (Kealhofer) 13

Table 3: Characteristics KMV Model 14

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III

List of Figures

Figure 1: Three Pillars 2

Figure 2: Today′s Best-Practice Industry Models 4

Figure 3: Black & Scholes (1973) 15

Figure 4: Distance to default (I) 16

Figure 5: Distance to default (II) 16

Figure 6: EDF - DD 10

Figure 7: Comparison of survey results 10

Figure 8: Survey 2002 12

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IV

List of Abbreviations

BCBS

= Basel Committee on Banking Supervision

CFF

= Credit Conversion Factor

DD =

Distance-to-default

DPT

= Default Point

EaD

= Exposure at Default

EDF

= Expected Default Frequency

EL =

Expected

loss

IRB

= Internal Ratings Based Approach

LGD

= Loss Given Default

LTD

= long-term debt

PD

= Probability of Default

STD

= short-term debt

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1

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

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.

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2

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 introduc-

tion 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

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.

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3

· 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 statis-

tical 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) de-

termined 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 commit-

ments likely to be drawn prior to default.16

COMM17

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

1

R

18

and refers further to the expectation of the severity: SEV

LGD19

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; = 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,...,

nt

. The number of defaulted bonds in year

t

,

t=1,...,

T

is denoted with

nt

.

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

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4

A model that takes the correlation between recoveries and default risk into account is

Moody′s KMV LossCalcTM.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:

EL EAD * LGD * DP 21

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.

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

20 Cp. Moody′s K M V, 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.

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5

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 estimat-

ing 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

,

xN

ce

29

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

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 K M V, Methodology Frequently Asked Questions.

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6

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

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.

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7

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 market-

to-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 follows 47

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 K M V, EDF Overview.

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

43 Cp. Moody′s K M V, 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.

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8

V

f V , , K, c, r 48

and thus the asset value VA as

V , , K, c, r . 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.

A

51

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

48 Denotation: VE = value of equity; A 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 K M V, 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 C

52 Cp. Moody′s K M V, EDF Overview.

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

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9

An example of a less risky (Firm I) and a more risky company (Firm II) should exem-

plify the resulting differences in DD, according to the formula in figure 3.54

Firm I

Firm II

Short-term debt

0,8

Short-term debt

1

Long-term debt

3

Long-term debt

3

Default point

3,4

Default point

3,5

Market value of

12,6 bn.

Market value of

12,6 bn.

assets

assets

Asset volatility

0,15

Asset volatility

0,15

DD

1

DD

1

12,6

3

2 0,8

12,6

3

2 1

4,9

4,2

0,15 12,6

0,17 12,2

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

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.

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10

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

Figure 6: EDF - DD

smaller.

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:

0,04

0,4%

40 59

.

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 EDFTM is higher as the survey

Figure 7: Comparison of Survey Results with Other Probability of

Default Measures

results in the good rating classes.

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

57 Cp. Moody′s K M V, 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.

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11

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 LossCalcTM 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.

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

62 Cp. Moody′s K M V, LOSSCALC V2: DYNAMIC PREDICTION OF LGD, p. 1.

63 Cp. Moody′s K M V, 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.

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12

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

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.

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13

Attachment

Table 1

68

"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 2

69

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).

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.

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14

Table 3

70

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.

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

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15

Figure 3

71

Black & Scholes (1973)

, ,

xN

ce

ln /

1

2

t

t

ln /

1

2

t

t

Denotation:

N(d) =

Cumulative normal density function

w

=

Value of the option

x

=

Stock price

t

=

Time

t*

=

Maturity date of the option

c

=

Exercise price

r

=

Interest rate

v2

=

Variance rate

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

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16

Figure 4

72

Distance-to-default (I)

DD

Denotation:

STD =

Short-term

debt

LTD =

Long-term

DPT

=

Default point = STD + ½ LTD

DD

=

Distance-to-default

E(V1) =

Expected asset value in 1 year

A

Volatility of asset return

Figure 5

73

Distance-to-default (II)

/

µ

1

DD

2 T

T

Assumption:

· Lognormality assumption of asset values

(Standard geometric Brownian motion)

Denotation:

V0

=

Current market value of assets

DPTT =

Default point at time horizon T

=

expected net return on assets

=

Annualized asset volatility

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.

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17

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