Abdelkader Derbali
email: derbaliabdelkader@outlook.fr

The credit portfolio management by structural models: A theoretical analysis

Keywords: Risk management, Credit risk, Default probability, Structural models, KMV model

The purpose of this paper is to study the credit portfolio management by the structural models (Moody's KMV model and CreditMetrics model) also defined by the models of the value of the firm. The development of this type of models is based on a theoretical basis developed by several researchers. The evolution of their default frequencies and the size of the loan portfolio are expressed as functions of macroeconomic and microeconomic conditions as well as unobservable credit risk factors, which explained by other factors. We develop two sections to explain the different characteristics of those two models. The purpose of all its models is to express the default probability of credit portfolio.

Introduction

The problem of evaluation of the failure probability of any borrower was the center of the bankers as soon as they began to lend some money. The quantitative modeling of the credit risk for a debtor is rather recent in fact. Besides, the modeling of the credit risk associated with instruments of a portfolio of credit such as, the loans, the pledges, the guarantees and the by-products (who constitute a recent concept).

A certain number of models were developed, including at the same time the applications of property developed for the internal custom by the financial institutions, and the applications intended for the sale or for the distribution (Hickman and Koyluoglu, 1999).

The big financial institutions recognize his necessity, but there is a variety of approaches and rival methods. There are two types of models of credit portfolio in the course of use at present [START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF]:  Moody's KMV model (Portfolio Model).

 CreditMetrics model by JPMorgan.

The main idea for this study is to answer the question follows: How the default probability is defined by the credit portfolio models?

Then, the organization of this paper is as follows. In section 2, we present the fundamental of structural models and we define the forces and the weaknesses of each model.

We provide the presentation of the Moody's KMV model in section 3. The section 4 is considered to present the development of the CreditMetrics model by JPMorgan. Section 5 concludes.

The Structural Models

The structural models of management of credit portfolio were presented by [START_REF] Merton | On the pricing of corporate debts: the risk structure of interest rates[END_REF] and then, developed by Leland (1994), Leland and Toft (1996), Anderson and Sundaresan (1996) and [START_REF] Jarrow | Credit market equilibrium theory and evidence: Revisiting the structural versus reduced form credit risk model debate[END_REF]. The characteristics to define a structural model are given by two conditions:

 The process of management of the assets of the company has to be known on the market in which this one operates.

 The structure of the liabilities of the company has to be known by all the actors operating on the market of this one.

In the practice, to examine the models of management of credit portfolio, it is necessary to employ parameters estimated implicitly because the values of the assets of the company are not observable. Nevertheless, the majority of the empirical evidence does not retain the structural models. The implicit prices obtained from the structural models does not seem to match the structure of maturity of the efficiencies on the assets of the company (Eom et al., 2004;Ericsson and Reneby, 2005;Jarrow et al., 2003;Schaefer and Strebulaev, 2008;Li and Wong, 2008;[START_REF] Jarrow | Credit market equilibrium theory and evidence: Revisiting the structural versus reduced form credit risk model debate[END_REF] and to allow the forecasts of defect of the borrowers (Patel and Pereira, 2007;Bharath and Shumway, 2008).

The analysis of the model of [START_REF] Merton | On the pricing of corporate debts: the risk structure of interest rates[END_REF] finds that this one supposes that the value of the firm follows a process of distribution and the defect occurs when the value of the firm falls below the nominal value of the debt on the date of maturity of this one. In this respect, this model serves to determine a threshold of defect.

The development of Merton's model is made by adding the other variables such as; the interest rate (Longstaff and Schwartz, 1995), the optimal permanent capital (Leland and Toft, 1996), the variable time of the threshold of default (Collin-Dufresne and Goldstein, 2001), the unfinished accounting information (Duffie and Lando, 2001) and the risk of the events of defect (Driessen, 2005).

The structural models are based on the theory of the options and the structure of the capital of the company [START_REF] Hamisultane | Modèles de gestion du risque de crédit[END_REF]. In this aligned, the bankruptcy of a company took place when the value of assets is situated below the value of its debt. The structural models or the models of the value of the firm are based on the approach of [START_REF] Merton | On the pricing of corporate debts: the risk structure of interest rates[END_REF] which supposes that the failure of a company appears in case the market value of its assets is lower than a certain threshold of its debts.

Generally, the models of credit portfolio management resting on the approach of Merton are the model KMV (Kealhofer, McQuown and Vasicek) of Moody and the CreditMetrics model of JPMorgan (1997). The distinction between both structural models was described in the table below.

Table 1: The comparison between the KMV model and the CreditMetrics model

The KMV approach

The CreditMetrics approach  The conduct of the value of the asset.  Companies are decomposed into systematic components and that no-systematic.  The systematic risk is based on the industry and the country of debtor.  The correlation of defect ensues from the correlation of assets.

 The indication of own capital.  Companies are decomposed systematic components and that no-systematic.  The systematic risk is based on the industry and the country of debtor and can be sensitive to the size of the asset.  The correlation of the defect ensues from the correlation of the efficiencies on own capital. Source: Smithson (2003) The structural models are also called models of the asset volatility. The Structural aspect of the models comes because there is a historical story behind by default that is something manages to start by default. The structural models are rooted in the knowledge of Merton. In Merton's model, the correlation of defect has to be a function of correlation of assets. The estimation of a structural model requires the implementation of the market value of the assets of the company and its volatility.

In the practice, the value of assets and their volatility are not observable for the most part of companies. The structural models lean strongly on the existence of assets quoted on the stock exchange so that we can estimate the necessary parameters.

The KMV model

The KMV model of credit portfolio management was elaborated for the first time in 1993. This model allowed the development of several models of quantification of the credit risk: Credit Monitor, Credit Edge and Private Firm Model for the individual credit risk and Portfolio Manager for the credit risk of a portfolio.

The model KMV rests bases on the notion of default distance which is calculated by basing itself on the barrier which engages the defect. As soon as, the distance in the defect is calculated, it transformed into the probability of failure (Expected Default Frequency: EDF).

The KMV model which was developed by the Moody's-KMV company is based on the theory of the prices of Merton options. It is about an abstract frame used to estimate the default probability of a company. The KMV model supposes that the company is in situation of defect when the value of its asset is less than the value of its debts. The Figure 1 explains the relation among the estimated own capital and the value of the asset. According to Merton's basic idea, in the KMV model the value of the own capital of the company is considered as being an option to buy. So, the value of the asset is considered as being the underlying asset and the debt represents the price of exercise (Chen et al., 2010).

Figure 1:

The relation between the market value of the assets of the company and the value of the debt (Merton,1974).

In Figure 1, V A indicates the initial investment of the shareholders of the company; X indicates the point of default which corresponds to the sum of the long-term debt and half of the current liabilities. When the value of assets (V A ) is superior to the debt (X), the shareholders will choose to gain profits staying after payment of the debts (V A -X) and these will be chosen by default, what is shaped with a net value raised in the Figure 1. In this case, the investor executes the option to buy.

So, if the value of assets is lower than the debt (V A < X), the shareholders will choose by default the transfer of the active total for the benefit of the creditors, what is coherent with a constant value of own capital indicated in the Figure 1, and it means that the option to buy is not executed (Caouandte et al., 1998;Kealhofer and Bohn, 2001;Saunders and Allen, 2002;Bohn and Crosbie, 2003).

Generally speaking, the shareholders receive Max (V A -X, 0) in the date of maturity T. According to Merton's model, the evolution of the market value of the assets of the company follows a process of geometrical distribution of the following shape:

 A At A

dV μ dt σ dW V

The market value of the assets of the company (

V A ) 0 V1 X V2
The value of the debt

-VA
Where, denotes the process of Wiener Standard is, is the average of the efficiency of assets and is the standard deviation of the efficiency on assets. The market value of the company is given by basing itself on the purchase price of a European option to buy supplies by Black and Scholes (1973).

    12   rT EA V V N d e XN d
Where, N(.) indicate the function of distribution of the normal law with (Huang and Yu, 2010):

2 2 1 1 () 11 2 () 2                  A A A A AA V ln r σT V X d ln r σT X σ T σ T 2 21 1 () 2           A A A A V ln r σT X d d

σT σT

In the KMV model, there is a hypothesis which rests on the structure of the capital of the company. So, this capital has to consist only by actions, current liabilities and in the long term and convertible prices. Really, the value of the company V A and the volatility of assets are not observable (Hull, 1997;Chen et al., 2010). We are going to deduct these two values by using the values of the options V E . So land us note that: ( , , , , )

 E A A V f V σ X c r ( , , , , )  E A A σ g V σ X c r
Where, c is the coupon paid on the long-term debt, r is the interest rate without the risk and is the volatility of share prices.

By applying the Lemma of Itô to these two functions and by arranging the terms we obtain:

()    AE EA EA VV σσ VV With: 1 ()    E A V N d V
who is deducted from the equation which measures the value of the V E which is defined by the following expression:

    12   rT EA V V N d e XN d Thus, 1 ()     A EA E V N d σσ V
Further to this transformation, we obtain a system of equation to two unknowns and :

    12 0     rT AE V N d e XN d V   1 0  E E A A σ V V N d σ
If the expressions of and are determined, then, we can arrive at the writing of the following formulation of the distance of defect (DD):

2 1 ln 2         A A A V μ σ X DD σT
According to the KMV model the distance of defect is defined in the following way (Crosbie and Bohn, 2003):

  A AA VX DD σV
From the distance of defect, we can deduct the value of the default probability as follows:

    2 1 2 () X                    A A KMV A A V ln μ σ T X P Prob V T N N DD σT
Then, we can obtain the frequency planned by default (Expected Default Frequency: EDF) such as:

()  EDF N DD
However, the default probability does not correspond to the normal law. KMV Company tries to obtain the empirical value of the EDF rather than the theoretical value of the models (Zheng, 2005). Fortunately, KMV Company possesses an enormous base of historical data concerning the default of the companies. By basing itself on these data KMV defined tables which associate with the various possible values of the distance of default (DD) on a temporal horizon considered a default probability definite and noticed empirically.

So, to protect itself against the risk which results from potential losses bound to the evolutions of the portfolio, Kealhofer, McQuown and Vasicek (1993) based on the determination of a random size L relative to the losses of the portfolio which is defined in a general way and on a horizon H as follows:

 HH ND L V V
Where,

H ND

V indicates the value of the portfolio H in the absence of the losses and indicates the market value of the portfolio H. The development follows by KMV shows us that the distribution of L can be approached by an inverse normal distribution.

Table 2:

The forces and the weaknesses relative to the KMV model

The forces

The weaknesses  The default probability is connected with the information of the market.  Contrary to CreditMetrics models and CreditRisk+ models the debtors are specific. We can distinguish them by basing itself on their default probability, on their own structure of capital and on their own assets.



The threshold of defect is determined in an empirical way.

 A hypothesis which is not realistic because she supposes that the debt of the company consists by bonds with zero-coupon and shares.  KMV supposes that the price of assets follows one moment Geometric Brownian. This modeling by a continuous process excludes all the early defaults.  This method is difficult because it depend a several data which are in most of the time unobservable or with difficulty accessible.  The interest rate is supposed constant.

Source: [START_REF] Hamisultane | Modèles de gestion du risque de crédit[END_REF]

The CreditMetrics model

CreditMetrics was thrown for the first time in 1997 by JP Morgan's bank.

CreditMetrics is considered as being an evaluation tool, for a portfolio, its variance of the values provoked by the changes of the quality of credit of the transmitter of the bonds (the credit migration) and leaves the defect of the counterpart. Unlike the approaches developed by the other models of management of a portfolio of credit, the probability of default in CreditMetrics is given by rating agencies for the big companies and by methods of scoring and mapping for small and medium-sized firms (Paleologo et al., 2010).

CreditMetrics belongs to the structural models since it rests on the model of [START_REF] Merton | On the pricing of corporate debts: the risk structure of interest rates[END_REF] for the definition of the thresholds of the migration of credit [START_REF] Jarrow | Credit market equilibrium theory and evidence: Revisiting the structural versus reduced form credit risk model debate[END_REF]. According to [START_REF] Hamisultane | Modèles de gestion du risque de crédit[END_REF], CreditMetrics makes it possible to calculate CreditVaR of a portfolio.

The methodology of this model is based on the probability of moving of a quality of credit to the other in a given horizon of time (analysis of the migration of credit). The calculation of CreditVaR by CreditMetrics rests on the four stages following [START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF][START_REF] Hamisultane | Modèles de gestion du risque de crédit[END_REF]:

 Determination of the risk isolated from each credit of the portfolio.

 The construction of the matrix of the probabilities of transition from a notation to another.

 The valuation of the assets of the portfolio according to the scenarios of transition from a notation to the other one.

 The calculation of CreditVaR.

The evaluation of a portfolio Value-at-Risk due to the credit (CreditVaR) by

CreditMetrics is given the following Figure 2 [START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF]:

The portfolio in an obligation

According to [START_REF] Hamisultane | Modèles de gestion du risque de crédit[END_REF], the system of rating used by CreditMetrics is the one rating agency. So, the broadcasting issuers of debt securities are noted according to a ladder of seven categories going from AAA to CCC according to the financial solidity of every company [START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF]. The notation AAA is tuned to the healthy companies financially whereas those who are characterized by a bad financial situation are noted by CCC.

The notations offered by the agencies of rating are regularly published. These notations present information relative to the broadcasting issuers of debt securities. The agencies of rating include these notations in indicating tables, either the rate of historic default of broadcasting issuers according to their notation on a horizon of well determined time, or the evolutions of these notations in the time. These tables recapitulating the notations tuned to the broadcasting issuers of debt securities are defined by "the matrices of transition".

The matrices of annual transition summarize all the changes of notation, on a horizon of time of one year, of a sand of broadcasting issuers is presented as follows: According to [START_REF] Grundke | Importance sampling for integrated market and credit portfolio models[END_REF], this table must be carefully analyzed. So, by taking as an example the line corresponding to the BBB rating presented in the table above, we can deduct the probability of default as follows: After a period of one year, and settling on the asset of initial notation BBB, we can deduct that the probability that this active rest BBB after a period of one year is 86,93 %, that to become AAA is 0,02 % and that to be lacking is 0,18 %.

The use of this model is based on three main hypotheses (JP Morgan and Co. Inc, 1997;Glasserman and Li, 2005;[START_REF] Hamisultane | Modèles de gestion du risque de crédit[END_REF][START_REF] Grundke | Importance sampling for integrated market and credit portfolio models[END_REF][START_REF] Figlewski | Modeling the effect of macroeconomic factors on corporate default and credit rating transitions[END_REF]:

 The absence of multiple transitions: for a horizon of time given the number of transitions is in most of a single transition.

 The stability of the matrix of transition in time: for every class of notation, two companies in different sectors or in different countries have the same probability to migrate from a notation to the other one.

 The matrix of transition is of type Markov: for period given the probability to migrate of a class of notation in another class is independent from what took place for the last periods. These hypotheses are emitted for the simplification of the calculations of the matrix of transition for the posterior periods.

CreditMetrics determines the current value of the bond by using the curve of the rates with zero coupons to proceed with the calculations of CreditVaR. In that case, the transmitter of debt securities is not in situation of bankruptcy. By continuing in the same context of analysis, that is the use of the notation BBB as the example, we can use the table of the Forward rates following: We suppose in our case which a noted transmitter BBB has emitted a Bond for 100 Euro over 4 years with a rate without annual risk of 6 %. The current value of the bond is given by the equation below:

        234 6 6 6 106 6
107.55 1 4.1 1 4.67 1 5.25 1 5.63

          V % % % %
By basing itself on the formula above, being able to us determine the various possible values of fire of type BBB according to his possible migrations towards other notations [START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF][START_REF] Hamisultane | Modèles de gestion du risque de crédit[END_REF]. The possible values of a bond rated BBB according to the possible migrations are presented in the table 5.

In this alignment, we suggest taking into account the probability of migration of a bond rated initially by BB. These probabilities are given by the following table: By basing itself on the model of [START_REF] Merton | On the pricing of corporate debts: the risk structure of interest rates[END_REF], we can suppose that the efficiency on a bond modeled as follows:

 r μ σε

With, a term of error is such as ~(0,1) ε N , is the average efficiency on the bond and is the standard deviation of the efficiencies of this bond. Then, the default probability of an issuer of the bond is given by the following expression:

           Def Def

Pr default Pr r Z Pr μ σε Z

Thus,

          Def Def Pr default Pr r Z Pr σε Z If                Φ Def Def

ZZ Pr default ε σσ

Where, indicates the cumulative function of the normal law.

By using the table above, we can establish the table according to who summarizes the distribution of the probability of migration affected in conformance with BB rating: With, 1     Φ AA Z σ represent the probability so that the bond of BB rating can pass in the notation AAA and Z AA indicates the threshold from which BB passes to AAA.

The transformation graphic of the data above is presented as follows:

Z% σ σ

The values of the other thresholds are calculated according to whom corresponds itself aside type of the normal distribution of the random on the assets of the notation BB [START_REF] Gupton | CreditMetricsTM -Technical Document[END_REF][START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF]Nickell et al., 2000;Bangia et al., 2002;Albanese and Chen, 2003;Albanese et al., 2003;Rosch, 2005;Feng et al., 2008).

We suppose now, that a second issuer presents a rating A where the random on assets follow a normal distribution with a parameter . In that case, the values of thresholds relative for two bands who rated BB and A are presented as follows: The last column of the table and the last line of this one represent the marginal probability for the entities BB and A which are equal to the sum of the joint probability by line or by the column. According to [START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF] these marginal probabilities correspond to the probability of migration of BB and of A taken individually. The variation of the portfolio of both bands is calculated for each of the joint probability (Brady and Bos, 2002;Brady and al., 2003).

The portfolio in several obligations

In case the portfolio consists further more than 2 bands calculates its joint probability will more be complicated. So, model CreditMetrics propose the use of the simulations of Monte Carlo and the decomposition of Cholesky to generate trajectories correlated to the bond and build the distribution of the values of the portfolio on certain horizon of time (Gouriéroux and Monfort, 1995;Fishmen, 1997;[START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF]Hamilton et al., 2002).

According to [START_REF] Hamisultane | Modèles de gestion du risque de crédit[END_REF]Feng et al. (2008), to generate trajectories correlated to the variables which follow a normal distribution N (µ, ∑). The determination of these trajectories requires the respect for the following five stages:

Stage 1: The regression of the random of the band on the sectorial indications. For example, in the case of three bands and two sectorial indications.

1, 1 1, , 1, , 1,     t X X t Y Y t t r α α I α I ν 2, 2 2, , 2, , 2,     t X X t Y Y t t r α α I α I ν 3, 3 3, 3, , 3,     t X

X t Y Y t t r α α I α I ν

To pass in the second stage it is necessary to estimate the various parameters of three models.

Stage 2: The calculation of the variances and the covariance's banding 2 bands i and j: And By using these two formulae, we can obtain the matrix of the variances-covariance's ∑.

The forces

The weaknesses  In the model CreditMetrics, both aspects of the credit risk are taken into account.

 The rating according to companies must be correct.  The interest rates are supposed constant.  The existence of a relation between the economic situation and the probability of defect. In that case, every economic cycle has to have matrices of transition appropriate for him.  The variability of the actions of a company can be used to deduct the variability of the price of the assets of the company. Source: [START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF], Nickell and al. (2000) and Bangia and al. (2002)

Conclusion

In this paper we developed a comparative theoretical approach's concerning the model of management of credit portfolio. Then, we studied the four mains models of credit portfolio management. In the financial literature those models are grouped by three types of credit portfolio models [START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF]. The structural models: there are two models of management of credit portfolio who are supplied in the literature: Moody's KMV model (Portfolio Model) and CreditMetrics model by JPMorgan.

The KMV model and Credit Portfolio View base their approach on the same empirical observation that default and migration probabilities vary over time. The KMV model adopts a microeconomic approach which relates the probability of default of any obligor, to the market value of its assets.

Figure 2 :

 2 Figure 2: The evaluation of a portfolio

Figure 3 :

 3 Figure 3: Generalization of the Merton model to include rating changes[START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF]

  and r' indicate respectively the random on the assets who are rated by BB and A and represent the joint density function by the Gaussian distribution which depends on the coefficient of correlation ρ. The joint density function of the Gaussian distribution of two variables X and Y is presented by the form below: According to Hamisultane (2008), for ρ = 20% the matrix of joint transition which considers the correlation banding both entities BB and A is the following one:

Stage 3 :Stage 5 :

 35 The decomposition of Cholesky of the matrix of the variances of the covariance's ∑ in the following way[START_REF] Hamisultane | Modèles de gestion du risque de crédit[END_REF]: the lower triangular matrix and A T transposed by the matrix A. Stage 4: The simulation of variables   , ~0,1 N  it Z . In fact, the existence of the bond to be feigned allows the existence of . The simulation of the values of the correlated variables et al. (2000), Nickell et al. (2000) and Bangia et al. (2002), the forces and the weaknesses of this model are presented in the following table:

Table 3 :

 3 Transition matrix: Probabilities of credit rating migrating from one rating quality to another, within 1 year

	Rating	AAA	AA	A	BBB	BB	B	CCC	Default
	AAA	90.81%	8.33%	0.68%	0.06%	0.12%	0.00%	0.00%	0.00%
	AA	0.70%	90.65%	7.79%	0.64%	0.06%	0.14%	0.02%	0.00%
	A	0.09%	2.27%	91.05%	5.52%	0.74%	0.26%	0.01%	0.06%
	BBB	0.02%	0.33%	5.95%	86.93%	5.30%	1.17%	0.12%	0.18%
	BB	0.02%	0.14%	0.67%	7.73%	80.53%	8.84%	1.00%	1.06%
	B	0.00%	0.11%	0.24%	0.43%	6.48%	83.46%	4.08%	5.20%
	CCC	0.22%	0.00%	0.22%	1.30%	2.38%	5.00%	64.85%	19.79%
	Default	0.00%	0.00%	0.00%	0.00%	0.00%	0.00	0.00%	100%

Source:

Standard & Poor's CreditWeek (1996)

Table 4 :

 4 The potential rating relative to the BBB rating

	Initial rating	Potential rating in a one year	Probability
		AAA	0.02%
		AA	0.33%
		A	5.95%
		BBB	86.93%
	BBB	BB	5.30%
		B	1.17%
		CCC	0.12%
		D	0.18%
		Total	100.00%

Source:

[START_REF] Grundke | Importance sampling for integrated market and credit portfolio models[END_REF]

Table 5 :

 5 One-year forward zero-curves for each credit rating (%)

	Category	Year 1	Year 2	Year 3	Year 4
	AAA	3.60	4.17	4.73	5.12
	AA	3.65	4.22	4.78	5.17
	A	3.72	4.32	4.93	5.32
	BBB	4.10	4.67	5.25	5.63
	BB	5.55	6.02	6.78	7.27
	B	6.05	7.02	8.03	8.52
	CCC	15.5	15.02	14.03	13.52
				Source: CreditMetrics, JP Morgan

Table 7 :

 7 Transition matrix based on actual rating changes

	Initial				Rating at year-end (%)
	Rating	AAA	AA	A	BBB	BB	B	CCC	Défaut
	AAA	90.81	8.33	0.68	0.06	0.12	0	0	0
	AA	0.70	90.65	7.79	0.64	0.06	0.14	0.02	0
	A	0.09	2.27	91.05	5.52	0.74	0.26	0.01	0.06
	BBB	0.02	0.33	5.95	86.93	5.30	1.17	0.12	0.18
	BB	0.03	0.14	0.67	7.73	80.53	8.84	1.00	1.06
	B	0	0.11	0.24	0.43	6.48	83.46	4.07	5.20
	CCC	0.22	0	0.22	1.30	2.38	11.24	64.86	19.79
						Source: Standard & Poor's CreditWeek (1996)

Table 8 :

 8 The distribution of the probability of migration of BB rating

	Rating	Probability from the transition	Probability according to the asset value
		matrix (%)	model
	AAA	0.03
	AA	0.14
	A	0.67
	BBB	7.73
	BB	80.53
	B	8.84
	CCC	1.00
	Default	1.06
			Source: Crouhy and al. (2000)

Table 9 :

 9 Transition probabilities and credit quality thresholds for BB and A rated obligors

		Rated-A obligor	Rated-BB obligor
	Rating in 1 year	Probabilities (%)	Thresholds:	Probabilities (%)	Thresholds :
	AAA	0.09	3.12	0.03	3.43
	AA	2.27	1.98	0.14	2.93
	A	91.05	-1.51	0.67	2.39
	BBB	5.52	-2.30	7.73	1.37
	BB	0.74	-2.72	80.53	-1.23
	B	0.26	-3.19	8.84	-2.04
	CCC	0.01	-3.24	1.00	-2.30
	Default	0.06		1.06
				Source: Crouhy and al. (2000)
	By taking into account the table above, we can calculate the probability of
	migration joined in the following way:

Table 10 :

 10 Joint rating probabilities (%) for BB and A rated obligors when correlation banding asset random is 20%

	Rating of	Rating of second company (A)
	first
	company	AAA	AA	A	BBB	BB	B	CCC	Default Total
	(BB)
	AAA	0.00	0.00	0.03	0.00	0.00	0.00	0.00	0.00	0.03
	AA	0.00	0.01	0.13	0.00	0.00	0.00	0.00	0.00	0.14
	A	0.00	0.04	0.61	0.01	0.00	0.00	0.00	0.00	0.67
	BBB	0.02	0.35	7.10	0.20	0.02	0.01	0.00	0.00	7.69
	BB	0.07	1.79	73.65	4.24	0.56	0.18	0.01	0.04	80.53
	B	0.00	0.08	7.80	0.79	0.13	0.05	0.00	0.01	8.87
	CCC	0.00	0.01	0.85	0.11	0.02	0.01	0.00	0.00	1.00
	Default	0.00	0.01	0.90	0.13	0.02	0.01	0.00	0.00	1.07
	Total	0.09	2.29	91.06	5.48	0.75	0.26	0.01	0.06	100
						Source: CreditMetrics, JP Morgan (Lucas, 1995)

Table 11 :

 11 The forces and the weaknesses relative to the CreditMetrics model

In case the company had a bankruptcy, the value of the bond is determined by using the average recovery ratio calculated by CreditMetrics on historical data (Carty and Lieberman, 1996;Gordy, 1998).

Further to the representative table of the various values of BBB according to the possible migrations, we can subtract the distribution of the variations of the price of the obligation in the following table: The analysis of this table shows that CreditVaR in 1 % (at a level of 99 % confidence) is equal to the last value of the variation of the value of the bond which corresponds to the notation CCC. Thus, CreditVaR is equal to -23.91.

The portfolio in two obligations

In the case of a portfolio consisted of two bands, the analysis is based on the level of correlation of the migrations. In fact, in a portfolio consisted of several assets the migrations of the various credits are correlated. CreditMetrics tries to estimate these correlations. As long, as there are no good data to be employed. In this case, CreditMetrics used the correlations between the values of the assets of the broadcasting issuers of the credits which are approached by the correlations among the equity prices of these broadcasting issuers to calculate the correlations between the migrations of the credits (Treacy and Carey, 2000;Altman and Rijken, 2004;Gordy and Howells, 2006;Xing et al., 2012).

According to Iscoe et al. (1999), to be able to divert the correlations of the migrations of the credits of the correlations of the values of assets, it is necessary to have a model linking the quality of a credit to the value of assets. The solution proposed by CreditMetrics is to use an extension of the model of [START_REF] Merton | On the pricing of corporate debts: the risk structure of interest rates[END_REF] which incorporates the migrations of the credits.


Abdelkader & Derbali

The credit portfolio management by the econometric models: A theoretical analysis

Keywords: Risk management, Credit risk, Default probability, Credit Portfolio View. JEL Classification: G13, G21, G28

This main idea of this paper is to examine theoretically the current model of credit portfolio management. We employ the credit portfolio view to examine the default probability measurement. The development of this type of model is based on a theoretical basis developed by several researchers. The evolution of their default frequencies and the size of the loan portfolio are expressed as functions of macroeconomic and microeconomic conditions as well as unobservable credit risk factors, which explained by other factors. We developed three sections to explain the different characteristics of this model. The purpose of this model is to assess the default probability of credit portfolio.

Introduction

The problem of evaluation of the failure probability of any borrower was the center of the bankers as soon as they began to lend some money. The quantitative modeling of the credit risk for a debtor is rather recent in fact. Besides, the modeling of the credit risk associated with instruments of a portfolio of credit such as, the loans, the pledges, the guarantees and the byproducts (who constitute a recent concept).

A certain number of models were developed, including at the same time the applications of property developed for the internal custom by the financial institutions, and the applications intended for the sale or for the distribution (Hickman and Koyluoglu, 1999).

The big financial institutions recognize his necessity, but there is a variety of approaches and rival methods. There are three types of models of credit portfolio in the course of use at present [START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF]: The main idea for this paper is to answer the question follows: How the default probability is defined by the credit portfolio models?


Then, the organization of this paper is as follows. In section 2, we define the econometric models and we define the main characteristics of the model Credit Portfolio View. We provide the forecast of the default rate in section 3. The section 4 is considered to present the conditional matrices of transition. We conclude in section 5.

The Econometric Models (CREDIT PORTFOLIO VIEW OF MACKINSEY)

Credit Portfolio View is a model with multiple factors which is used to feign the common conditional distribution of the default probability and migration for various groups of estimation and in different industries [START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF].

This model was developed by Wilson (1997) within McKinsey. The approach developed by this author bases itself on the hypothesis that the probability of defect and migration are connected to macroeconomic factors such as the level of the long-term interest rate, the growth rate of the GDP, the global unemployment rate, the exchange rates, the public spending, the savings.

Credit Portfolio View is based on the occasional observation which supposes that the default probability, as well as the probability of migration, is connected to economic cycles. When the economy is in situation of recession, then the cycles of credit are also lesser. If it is the opposite case (the economy is in situation of expansion) then the cycles of credit become stronger.

In other words the cycles of credit follow the tendency of economic cycles.

Because the state of the economy is widely driven by macroeconomic factors, Credit Portfolio View proposes a methodology to connect these macroeconomic factors to the probability of default and migration.

Provided that the data are available, this methodology can be applied in every country, in the different sectors and in the diverse classes of borrowers of the obligors who react differently within the economic cycle.

The way that a model Credit Portfolio View works is as follows (Smithson, 2003):

 Simulate the state of the economy.

 Adjust the rate of default to the state of the simulation of the economy.

 Attribute a probability of default for every debtor on the basis of the simulations of the state of the economy.

 The value of the individual transactions attributed to the debtors according to the probability of defect is determined on the basis of the simulations of the state of the economy.

 Calculate the loss of the portfolio by adding the results for all the transactions.

 Repeat all the stages quoted above certain number of times to map finally the distribution of the losses.

In

The forecast of the default rate

In the Credit Portfolio View model, the probabilities of default are modeled as being a Logit function. In this modeling the independent variable is a specific speculative index in every country and which depends on macroeconomic variables. The Logit function allows that the values of probability of default are included between 0 and 1 [START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF][START_REF] Hamisultane | Modèles de gestion du risque de crédit[END_REF] Where: , and are a coefficients to be estimated and is a term of error which is normally distributed and independent of .

. , , 1 1    jt jt Y P e , , 0 ,1 ,1, ,2 ,2, , , , ,  
In this frame, our objective is to resolve the system below:

, , 1 1

   jt jt Y P e , , 0 ,1 ,1, ,2 ,2, , , , ,  
   j t j j j t j j t j m j m t j t Y β β X β X β X ε , , , ,0 , ,1 , , 1 , ,2 , , 1 , 
    j i t j i j i j i t j i j i t j t X γ γ X γ X ω
Where is the vector of the innovations such as:

~(0, )      t t t ε E N ω And , ,               ε ε ω ω ε ω
Where, and Represent the matrices of correlation.

In case the parameters are estimated, then it is possible to feign the probability of default by basing itself on historical data. Credit Portfolio View uses tired matrices of transition of economic cycles.

The conditional matrices of transition

By basing itself on the matrices of transition in the economic cycles which are proposed by the Credit Portfolio View, we can determine the situation of the economy [START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF]. Noting in this respect that, the matrices of transition in the Credit Portfolio View are different to those of the matrices of migration in the CreditMetrics [START_REF] Hamisultane | Modèles de gestion du risque de crédit[END_REF].

Credit Portfolio View proposes a tool based on the following ratio:

,

jt P φSDP

Where, represent the probability of default feigned for date t and for the sector j and represent the historic default probability which is based on observed data. Credit Portfolio View suggests employing this ratio to adjust the probability of migration. So, the matrix of transition multi-period is given by the following formula: Where, M L indicates the matrix of transition in the case of a period of recession and M H indicates the matrix of transition in the case of a period of expansion.

We can simulate a lot of time the matrix of transition to determine the probability of default for any estimation and for any period. The methodology of Monte Carlo Simulation can be used to determine the distribution of the default probability for any period. The forces and the weaknesses relative to the Credit Portfolio View model are presented in Table 2. Source: [START_REF] Hamisultane | Modèles de gestion du risque de crédit[END_REF]

Conclusion

In this paper, we present a theoretical approach's concerning the model of management of credit portfolio by the Credit Portfolio View model. The Credit Portfolio View model proposes a methodology which links macroeconomics factors to default and migration probabilities. The calibration of this model necessitates reliable default data for each country, and possibly for each industry sector within each country. The probability of failure depends in these models of macroeconomic factors such as unemployment, the rate of increase GDP, the interest rate long-term.

  The structural models: there are two models of management of credit portfolio who are supplied in the literature: Moody's KMV model (Portfolio Model) and CreditMetrics model by JPMorgan.  The Macro-factors model (Econometric model): The Credit Portfolio View model introduces in 1998 by Mckinsey.  The actuarial models CSFP (Credit Suisse First Boston): this model (CreditRisk+) is developed in 1997.

  the model Credit Portfolio View of McKinsey, the historic rates of default for the various industries are described according to the macroeconomic variables specified by the user of the model: the approach McKinsey, the rates of defect are commanded by a sensibility in a sand of the factors of the systematic risk, or the specific factors to the company. The table below summarizes the main characteristics of the model of McKinsey (Smithson, 2003):

  the estimated factors of sensibility. The engine of the risk The adjustment of the ARMA model (Autoregressive Moving Average model) with the evolution of the macroeconomic factors. The shocks undergone by the system determine the standard deviation of the average of the rates of defect concerning the level of the segment.

  is in period of expansion.

Table 1 The main characteristics of the model Credit Portfolio View

 1

Table 2

 2 The forces and the weaknesses relative to the Credit Portfolio View model

	The forces		The weaknesses
		Credit			In the Credit
	portfolio	View	Portfolio View model,
	connects	the	we use macroeconomic
	probability	of	data which cannot be
	default and the	available for a country
	matrices	of	or a business sector.
	transition	with		This	model
	economic		determines only the
	indicators. In other	probability of default
	words,	the	of a country or a
	probability	of	business sector and not
	default is stronger	an issuer.
	in	period	of
	recession than in
	period	of
	expansion.

Assistant Professor, Higher Institute of Management of Sousse, Sousse University, Tunisia


Keywords: Risk management, Credit risk, Default probability, Structural models, KMV model, CreditRisk+, Credit Portfolio View. JEL Classification: G13, G21, G28

How the default probability is defined by the CreditRisk+ model?

Abdelkader Derbali

To cite this version:

Abdelkader Derbali. How the default probability is defined by the CreditRisk+ model?. 2018.

Introduction

The problem of evaluation of the failure probability of any borrower was the center of the bankers as soon as they began to lend some money. The quantitative modeling of the credit risk for a debtor is rather recent in fact. Besides, the modeling of the credit risk associated with instruments of a portfolio of credit such as, the loans, the pledges, the guarantees and the byproducts (who constitute a recent concept).

A certain number of models were developed, including at the same time the applications of property developed for the internal custom by the financial institutions, and the applications intended for the sale or for the distribution (Hickman and Koyluoglu, 1999).

The big financial institutions recognize his necessity, but there is a variety of approaches and rival methods. There are three types of models of credit portfolio in the course of use at present [START_REF] Crouhy | A comparative analysis of current credit risk models[END_REF]:  The actuarial models CSFP (Credit Suisse First Boston): this model (CreditRisk+) is developed in 1997.


Then, the organization of this paper is as follows. In section 2, we present the CreditRisk+ model and we define the forces and the weaknesses of this model. Finally, we conclude in section 3.

The Model CSFP: CREDIT RISK+ MARKET RISK

Since 1990s, Credit Suisse First Boston (CSFB) has developed new methods of risk management. In 1993, the credit Swiss Group launched, in parallel of an important project which aims at modernizing its credit risk management and by using the expertise of CSFB, new one management tool of the credit portfolio in the future. In December, 1996, Credit Suisse Group presented the CreditRisk+ model as being a model of the credit portfolio management.

The structural models present an inconvenience concerning the default. These models suppose that the default cannot have arisen by surprise because the market value of assets is supposed to follow a continuous process of distribution. In this aligned, a process of Fish was used in the actuarial models the purpose of which is to model the unpredictable character of the emergence of the default what is developed in the model CreditRisk+.

CreditRisk+ is a model with intensity is which presents no hypothesis on the causes of failure of a company. It is model statistical of the default of credit risk which makes no claim about the causes of the default. This approach is similar to that of the management of the market risk, in which no attempt is made model the causes of the movements of market prices. This model does not consider the consequences of a deterioration of the quality of the quality of the counterparty.

So, the number of failures in a credit portfolio during the given period justifies itself by a process of Fish. CreditRisk+ uses a methodology based on techniques and quantitative methods. The present model is based on an actuarial calculation to determine and present the distribution of the losses of a credit portfolio.

The CreditRisk+ presents four hypotheses:

 Every individual credit presents only two possible states: failures or no failures.

 The default probability of an individual credit is low.

 The default probability for a big group of borrowers is very low.

 The number of default over a period is independent from that of any other period.

By basing itself on these hypotheses, the probability distribution of the number X of defaults over a given period (one month or one year for example) can be represented by using the law of Fish of average µ and of standard deviation :

  !   n μ μe P X n n
Where, µ is the average of the number of default a year.

  A μP

Where, PA indicate the default probability of the obligor A.

The annual number of the defaults, n, is a stochastic variable of average µ and a standard deviation . According to CreditRisk+, the calculation of the distribution of the losses requires the use of an approach by bonds; that is issued in a portfolio are grouped and collected by edge of exposure.

The process of determination of the distribution of the losses of a portfolio is constituted by three stages:

 The determination of the generative function of probability for every bond.

 The diversion of the generative function of probability for the whole portfolio.

 The determination of the distribution of the losses for the whole portfolio.

The distribution of the losses of default for a portfolio is diverted in two stages as the watch represents it below: Until here, we suppose that the distribution of fish allows moving closer to the distribution of the number of the events of defect. Then, we should expect that the standard deviation of the default rate is approximately equal to the square root of the average.

In case of defect of an obligor, the counterparty incurs a loss equal to the quantity possessed by the obligor less a quantity of restoring. In CreditRisk+ the exposure for every obligor is adjusted by the rate planned by restoring, to calculate the loss of default. These adjusted exposures are exogenous in the model, and are independent of the market risk and minimize the risk.

To divert the distribution of loss for a diversified portfolio, the losses are divided into bands with the level of the exposure in every band.

To analyze the distribution of the resultant losses of the whole portfolio, presenting us the default probability expressed by the function defined in terms of variables auxiliary z by respecting itself the following approach of the formulation of the generative function:

We considered X a whole and positive random variable. The generative function of X is the whole series:

  0 ()     k k

G z P X k z

Where P(X=k) is the probability that the random variable X takes the value k. to obtain P(X=k) from the generative function G(z), we use the following formula:

  1 (0) !  k k dG P X k k dz
In that case, the generative function associated among default X arisen among all the bonds of a portfolio is given by the expression below:

        00 . . 1 !          exp n μ nn nn μe

F z P X n z z μ z n

This function can be written as follows:

  ()   A A F z F z
Where indicate the generative function of a portfolio constituted by a single bond of the issuer A.

So, every portfolio consists of m identical bond of exposure of indications j (j = 1, 2, m). In that case, the inputs of the model to be developed are: the individual exposure L and the probability of default for the issuer (debtor) A. Then, the loss hoped for the debtor A is expressed as follows:

 A A A λ L * P  A A λ ε L
The expression above is obtained when the expected loss is expressed in units of L. So, the expected loss for the bond j is given then as follows:

  jA ε ε
In this perspective, the expected number of defects for each of the indicated bond j is then given by:

         jj j AA j j j A ε εε μ
Thus, the number of waited defects total µ for them m bond is expressed as follows

11     mm j j jj j ε μ μ
The expression of the generative function of the included losses is obtained by:

    0     n n G z P Agregate losses n* L z   1 ()    m j j

G z G z

Thus:

  0 ( ).      j n j j j n

G z P V k z

Where represents the amount of the losses of the bond j and indicates the probability of the loss .

P z μ z μ ε

Then, the generative function of the included losses can be written in the following way:

          exp 1 ( )    Gz μ P z F P z
Where from, we can obtain the distribution of the losses of the total portfolio of an amount (n*L) as follows:

Land us note in that case that, can be calculated in continuous by basing itself on the following formula and under the hypothesis according to which µ is constant.

Where from we obtain:

The CreditRisk+ model considers that every sector is driven by a simple fundamental factor. This factor explains the variability of the rate of average defect measured for this sector. The fundamental factor influences the rate of defects planned in the concerned sector which is modeled by a random variable of average µ and of standard deviation indicated for every sector.

The standard deviation reflects the degree to which, in all the probability of default, the obligors in the portfolio are exposed are more or less that their levels of the average. By continuing this analysis, the model CreditRisk+ bases on the hypothesis that µ is constant. So, by basing itself on the distribution of Fish of parameter µ the probability of failures are underestimated. In that case, it is necessary to take into account the existence of an average number of variable failures.

In this alignment, the parameter µ is considered as being a stochastic variable and depends on characteristics of the sector. In fact, and according to the CreditRisk+ model, a sector is considered as being a sand of credits the rates of failure of which are subjected to the same influences. In the CreditRisk+ model, every portfolio is divided into sectors indicated by k with 1 ≤ k ≤ K.

    0 1 0 exp             m j j j ε AG μ exp          j j j n j n ε A n |
In particular, for every sector k, we introduce one random variable which represents the average number of defaults in this sector. The average number of the defects is equal in µ.

So, the hope of for the sector k is noted µ and its standard deviation is equal in . In this frame µ is calculated as follows:

() () () 1     k mk j k k j j ε μ
In the case that µ is no constant; the generative function of the number of defaults is given by:

    1    k k k F z F z And         1 0 00 ()         xz n k n xx

F z z P n defaults f x dx e f x dx

Where indicates the density of the variable .

The continuation of the calculations is conditioned by the presence of a nature of distribution given in . In the CreditRisk+ model, the choice is fixed to a distribution Gamma Г of average µ and of standard deviation .

Thus we obtain:

  1 ( 1) 0 ()        kk k x βα xz k α kk x ex F z e β Г α
Where the Gamma function written as follows:

  1 0      x α x Г α e X dx
For every sector k, we have two parameters of Gamma function to be estimated and .

Thus :

2 2  k k k μ α σ 2  k k k σ β μ
By substituting and by basing itself on the definition of the Gamma function, we obtain then:

  1 ( 1) 0 ()        kk k x βα xz k α kk x ex F z e β Г α       11 1 ( ) 1 1       Γ Γ kk kk k αα αα k k k k F z β α β z β β z
After this simplification, the generating function of the distribution of the probabilities of default for the sector K is given by the following expression:

 

F P z

Where the polynomial function is written as follows:

  () () () () () () 1 () () () () 1 () 1 1                        
 () () () 1 () 1 1 1                     k k j α k k k mk k j k k j kj p G z ε p z µ
In this respect, we can deduct the distribution of the losses of portfolios from the A n which is given by:

  0     n n n G z A Z
So, in case verify the following relation:

( ) ( ) ( ) ( )  ' G z A z G z B z
Where and are two polynomials of the following shape:

  0    r r A z a a z   0    s s B z b b z
Thus, the coefficients verify the relation of following recurrence:

A a A b n l A bn

This relation is applied knowing that verify the following condition:

() () () 1 () 0 () () 1 () 1 () () 1                        k j k j mk k kk j ' k j k k mk k j k k j kj p α εz µ Gz Gz ε p z μ
Generally, the CreditRisk+ model is based on mathematical techniques in the modeling of the distribution of the losses in the field of the banking activities and of the insurance. The behavior of common default of the borrowers is incorporated by treating the rate of default as being a common random variable for multiple borrowers. So, the borrowers are assigned among the sectors among which each has a rate of average default and a volatility of rate of default. The volatility of rate of default is the standard deviation which would be observed on a portfolio of infinitely diversified homogeneous credit.

The forces and the weaknesses relative to the CreditRisk+ model are presented in the table below [START_REF] Hamisultane | Modèles de gestion du risque de crédit[END_REF]:

Conclusion

In this study, we expose a theoretical approach's concerning the model of management of credit portfolio by the actuarial models CSFP (Credit Suisse First Boston). However, structural models are based on option theory and capital structure the company. On econometric models, they link the probability fault of the company to the state of the economy. The probability of failure depends in these models of macroeconomic factors such as unemployment, the rate of increase GDP, the interest rate long-term. Moreover, in the CreditRisk+ models, the probability of default varies over time.

  The structural models: there are two models of management of credit portfolio who are supplied in the literature: Moody's KMV model (Portfolio Model) and CreditMetrics model by JPMorgan.  The Macro-factors model (Econometric model): The Credit Portfolio View model introduces in 1998 by Mckinsey.

Figure 4

 4 Figure 4 CreditRisk+ risk measurement framework (Crouhy et al., 2000)

  average loss expressed in multiple of a standard exposure L, indicate the expected number of defaults which is a known value and indicate the exposure expressed in multiple of L in the band j.

Furthermore

  After the determination of the number of defaults in a portfolio, one goes in what follows to present the generating function of the losses incorporated in portfolio functions written is the following ;

  One can deduce the expression from the generating function which is written in the following way:



Table 1 The forces and the weaknesses relative to the CreditRisk+ model The forces

 1

	The weaknesses

Source:

[START_REF] Hamisultane | Modèles de gestion du risque de crédit[END_REF]


Luciano Da

F Costa

Peter F Drucker

Characterizing Germanium Junction Transistors

Keywords:

published or not. The documents may come

I. INTRODUCTION

It is not so often realized that most of modernity has been supported, and to some extent even defined, by electronics. Though the analogue is frequently said about computing, this important area itself relies strongly on electronics. As the development of modern electronics started with the transistor invention in 1947 [START_REF] Riordan | Crystal Fire -The Birth of the Information Age[END_REF], the above sentence could be rephrased as modernity, ultimately, being induced by the transistor. As a matter of fact, any cell phone or personal computer, not to say the myriad of other electronic devices and systems that permeate our lives, rely critically on millions of transistorsit remains an interesting question to estimate the number of transistors involved in the Internet. Though most of these devices employ silicon as basic semiconductor material, the first transistors employed germanium, which was dominant from 1947 to the mid 50's [START_REF] Riordan | Crystal Fire -The Birth of the Information Age[END_REF] and used until the 60's and even 70's. While MOSFET technology mostly predominates in modern analog electronics, * ldfcosta@gmail.com, luciano@if.sc.usp.br point contact, grown junction, alloy junction and surface barrier were employed from 1947 to mid 50's.

Both germanium (Ge) and silicon (Si) are semiconductor materials, characterized by increase of conductivity with temperature as free carriers are transferred from the valence to the conduction bands (e.g. [START_REF] Jones | A Practical Introduction to Electronic Circuits[END_REF][START_REF] Parker | Introductory Semiconductor Device Physics[END_REF][START_REF] Sze | Circuit Theory and Design[END_REF]). However, as summarized in Table I, these two semiconductor materials have quite distinct physical and electronic properties. In particular, the substantially larger mobility of germanium has great influence on the electronic properties of respectively derived transistors. Though this would represent, in principle, an advantage for germanium devices, there are some shortcomings potentially constraining the electronic applications of these devices relatively to silicon counterparts. These include the fact that more free electrons are available at room temperature in germanium than in silicon, implying considerably smaller current cut-off (I CBO ) for silicon devices. Germanium-based transistors are also characterized by smaller peak inverse voltage (P IV ) than silicon devices, averaging 350V and 1000V , respectively. In addition, the typical working temperature for germanium is only about 70 o for germanium, being much larger (about 160 o ) for silicon. Such issues contributed to the progressive shift from germanium to silicon as the basic semiconductor material adopted for industrialized transistors.

Except for I CBO , the germanium relative limitations are more critical only for given applications and circumstances, remaining a viable choice for many important usages, such as in many small signal and low power linear circuits, especially when special attention is given to temperature management and circuit design. So, germanium remains a potentially interesting choice in transistor electronics. Some important issues that could impact on the eventual use of germanium devices in linear electronics concern their electronic parameters (e.g. [START_REF] Jaeger | Microelectronic Circuit Design[END_REF][START_REF] Boylestad | Electronic Devices and Circuit Theory[END_REF][START_REF] Sedra | Microelectronic circuits[END_REF][START_REF] Stewart | Circuit Theory and Design[END_REF]), such as the current gain β and output resistance R o , as well as frequency features such as f T . Relatively little can be found in the literature regarding the more systematic characterization of real-world germanium devices. Actually, even the available data is often of relatively limited assistance because both β and R o tend to vary largely with I C and V C during normal circuit operation, while the given parameter values are typically limited to averages or specific to given operation points. A substantially more comprehensive characterization of the electronic properties of a transistor can be achieved by considering the whole region of operation in the V C ×I C characteristic space, where V C and I C stand for the collector voltage and current, respectively. In addition, even when taken from the very same lot, transistors tend to present largely varying parameters, implying several devices to be considered so as to achieve statistical significance.

The fact that germanium transistors have some potentially interesting physical features, especially higher mobilities, while their other limitations do not completely undermine their applications in many practical circuits and circumstances, entails the interesting prospect of verifying, in a more systematic and comprehensive way, how germanium transistors behave electronically. In particular, it would be interesting to compare small signal germanium and silicon devices, so as to try to identify their respective relative main pros and cons. These issues motivate the present work. This is a timely endeavor because, with the with the advances in instrumentation and information technology, more systematic, stable and comprehensive approaches to transistor characterization have become available. In particular, we resource to the Early effect-related method reported in [START_REF] Da | Characterizing complementary bipolar junction transistors by Early modelling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF], as well as the systematic approach to characterization of transistors described in [START_REF] Da | Negative feedback, linearity and parameter invariance in linear electronics[END_REF].

The application of these methodologies, while considering new old stock germanium junction transistors, led to several remarkable results, which are presented and discussed in this work. Of particular relevance is the fact that the distribution of the devices resulted much more definite and organized than that obtained in the more traditional and electronically intuitive space defined by the current gain β and the collector, or output resistance R o . This important result paves the way to more systematic characterization studies aimed at mapping several types of amplifying devices in the Early space, so as to assist the analysis and design of electronic devices and circuits.

This article starts by revising the theoreticexperimental-numeric Early characterization approach [START_REF] Da | Characterizing complementary bipolar junction transistors by Early modelling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] and proceeds by presenting and discussing the results obtained with respect to three types of germanium junction transistors as well as a comparison with results previously [START_REF] Da | Characterizing complementary bipolar junction transistors by Early modelling, image analysis, and pattern recognition[END_REF] obtained for 7 complementary types of silicon transistors.

II. EARLY MODELING AND CHARACTERIZATION

In 1952, an interesting electronic effect taking place in junction semiconductors was reported by J. M. Early, in which the width of the charge carrier portion of the base varies with the base-collector voltage. In typical transistor configurations, this implies that, for a fixed current base I B , the collector voltage V C will not be zero for I C = 0, but rather would take a negative value V a (the Early voltage), were it possible to cross over to negative values of V C . This is illustrated in Figure 1, which also shows the chosen region of operation (defined by 0 ≤ I C ≤ C,max and 0 ≤ V C ≤ V C,max ) and the load line defined by a chosen load resistance L and V C,max considering a simplified common emitter configuration in which the load is attached between the voltage supply V CC and the collector. It should be taken into account that, for simplicity's sake, all voltages and currents for the PNP devices are taken in absolute values in this work.

The angle β has been experimentally found [START_REF] Da | Characterizing complementary bipolar junction transistors by Early modelling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF], at least for the devices considered in that work, to be directly proportional to the base current, i.e. θ = sI B . The Early voltage V a and the proportionality parameter s constitute the two parameters of the Early modeling. This linear relationship turns out to be of great importance for deriving and applying the Early model, as it considerably simplifies the respective calculations and contributes to the simplicity of the approach and for the derivation of several analytical relationships. Observe that the Early model is also special in the sense that its two just mentioned parameters do not vary with either I C or V C , as it happens with the large majority of the more traditionally adopted junction transistors models. The more commonly adopted parameters known as current gain (β) and output resistance (R o ), also illustrated in Figure 1, are defined (e.g. [START_REF] Stewart | Circuit Theory and Design[END_REF] respectively as:

β = ∂I C ∂I B V C (1)
R o = ∂V C ∂I C I B (2)
In this more traditional parametrization, we have that both β and R o are functions of the two collector variables, i.e.

β = β(V C , I C ) and R o = R o ( ṼC , Ĩ C )
. So, at a given point ( ṼC , ĨC ), β needs to be approximated as β ≈ ∆I C ( ṼC , ĨB )/∆I B ( ṼC , ĨB ), while V a (and s) remains constant along the whole (V C , I C ) space. Still, β and R o are also required, because they correspond to more intuitive electronic properties of the transistors. Fortunately, it is possible to derive relationships [START_REF] Da | Characterizing complementary bipolar junction transistors by Early modelling, image analysis, and pattern recognition[END_REF] between the Early parameters and the values of β and R o , both averaged along the region of operation 0 ≤ I C ≤ C,max and 0 ≤ V C ≤ V C,max , which are given [START_REF] Da | Characterizing complementary bipolar junction transistors by Early modelling, image analysis, and pattern recognition[END_REF] as where V C,max is the maximum collector voltage considered (in this work, V C,max = 8V ), I C,min is the minimum collector current needed to be taken into account so as to avoid divergence in Equation 4 (here, I C,min = 1mA), and I C,max is the maximum considered collector voltage (henceforth taken as I C,min = 15mA). In this way, the relative advantages of the two parametrizations can be combined.

The Early parameters V a and s of a real-world transistor can be numerically estimated by applying the following three steps [START_REF] Da | Characterizing complementary bipolar junction transistors by Early modelling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF]: (i) the values of V C , I C , V B and I B are obtained experimentally by scanning the device with a succession of V CC values (note that other schemes can be used); (ii) a Hough transform accumulation (or voting) scheme is employed to identify the point in the (V C , I C ) space where the isolines (obtained by linear regression) intercept, leading to the estimation of V a ; and (iii) linear regression is applied in order to obtain s from the parameters I B and θ, the latter corresponding to the tangent of the angular coefficient of the respective isolines.

The above outlined methodology has been applied with encouraging success for the experimental characterization of BJT NPN-PNP complementary pairs [START_REF] Da | Characterizing complementary bipolar junction transistors by Early modelling, image analysis, and pattern recognition[END_REF], yielding several interesting results including the identification of two almost non-overlapping groups defined respectively by these two types of devices, each with well-defined specific electronic properties. In addition, all the considered devices were found to populate a relatively narrow curved band in the Early space, with β varying from 100 to 400. The main distinction between the PNP and NPN groups was that the former presents intrinsic larger parameter variation, as well as smaller magnitudes of V a and larger values of s . A prototypical Early space was outlined in [START_REF] Da | Characterizing complementary bipolar junction transistors by Early modelling, image analysis, and pattern recognition[END_REF], which provides a reference for comparison with other families or types of transistors and other amplifying devices. In the current work, the group of Germanium semiconductors is included into that Early prototype space.

A. Results for Germanium Devices

The experimental data considered in the present work derives from three types of small signal germanium junction transistors, 2 NPN and 1 PNP. All used germanium transistors are of alloy junction type. Ten samples of each were selected from homogeneous respective new old stocks. The acquisition, pre-processing and analysis of the respectively derived signals followed the methodology described in [START_REF] Da | Characterizing complementary bipolar junction transistors by Early modelling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF].

Figure 2 illustrates one of the obtained sets of I B -indexed isolines as acquired, without any smoothing or pre-processing, which indicates a good signal to noise level. Observe the increasing slope of the isolines as I B goes from 0 (bottom) to I B,max (top), which is an intrinsic characteristic of the Early effect. Next, each of

β = s 6VC,max 2I 2 C,max ln Va -VC,max Va + 3V 2 C,max -6VaVC,max (3)
Ro = V 2 C,max -VC,maxVa VC,maxIC,max ln IC,max IC,min (4)
these sets of isolines is slightly smoothed through an average filter (11 points wide) and resampled so as to have the same number of isolines in all cases [START_REF] Da | Characterizing complementary bipolar junction transistors by Early modelling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF].

FIG. 2: Example of isolines obtained for one of the NPN germanium transistors. These isolines are shown as acquired, without smoothing or resampling. The noise has been verified to be in the order, or smaller, than the resolution of the 12-bit ADCs, as a consequence of special care taken in the development of the customized acquisition system.

The Hough transform-inspired voting (or accumulating) scheme used for numeric estimation of V a typically resulted in sharp, well-defined peaks such as that illustrated in Figure 3. Observe that only a region of the (V a , s) space, corresponding to where the I B isolines tend to intersect, needs to be considered in the Hough mapping and voting. The V a coordinate (x-axis) of the peak is taken as the most likely value of V a for each transistor. Observe that the peak of isolines intersections in the (V a , s) space does not occur exactly on the V a axis, with a small offset in the order of one mV being obtained in the I C peak coordinates. The use of the Hough transform-inspired voting allows eventual isolines diverging too much from the others not to be considered.

Figure 4 presents the obtained distributions of germanium NPN and PNP junction transistors in the Early space (V a , s). The two NPN types, namely G1 and G2 resulted with very similar Early parameters, therefore exhibiting great overlap in the Early space in Figure 4. This means that these two types of NPN germanium transistors have very similar electronic properties, at least considering the adopted samples/configurations and non-reactive loads. The PNP group, identified as G3 in this figure, resulted with distinctively higher V a values (smaller V a magnitudes), and also larger values of the proportionality parameter s than the devices in the two NPN groups. The two groups have their centers of mass (averages) very near to the isoline defined for β = 130, which is shown by the dashed curve in this figure.

The scatterplot derived by the adopted Early methodology for the more traditional and electronically intuitive parameters β and R o is depicted in Figure 5. As in the Early mapping, the two NPN groups largely overlap one another, and are also less separated from the PNP than in that other space. Similar levels of parameter variations are observed in this space for any of the three types of germanium transistors. In addition, the PNP devices yielded smaller R o values.

B. Comparison with Silicon BJTs

It would be particularly interesting if the electronic characteristics obtained for germanium junction transistors could be compared with those of more modern silicon bipolar junction transistors -BJTs. Fortunately, this can be done by using recently obtained results [START_REF] Da | Characterizing complementary bipolar junction transistors by Early modelling, image analysis, and pattern recognition[END_REF] regarding the latter type of transistors. A total of 14 types of small signal BJTs (7 NPN and 7 PNP), each represented by 10 samples, were characterized in that work by using the same Early approach adopted here. Figure 6 presents the mean and dispersion (Mahalanobis ellipses [START_REF] Johnson | Applied multivariate statistical analysis[END_REF]) obtained for the four groups -NPN (Ge), PNP (Ge), NPN (Si) and PNP (Si) -in the (V a , s) Early space. Each ellipse was obtained from the respective covariance matrices by considering unit Mahalanobis distance [START_REF] Johnson | Applied multivariate statistical analysis[END_REF], specifying the same number of samples to comprised, in the average, inside each respective ellipse. The means of each group are marked by the "+" sign. The isolines corresponding to β = 250 and β = 130 are also represented by the dotted and dashed lines, respectively. The prototypical characteristic surfaces respectively induced in the (V C , I C ) transistor operation space are also shown in the separated plots, providing a direct graphic representation of the typically expected electronic properties for each of the considered groups.

Several remarkable results can be inferred from Figure 6. First, we have that the NPN-PNP groups obtained for germanium look like the silicon pairs scaled so as to have smaller V a magnitude and slightly higher values of s. Interestingly, the elongation of all groups tend to be mostly aligned with the tangent of the respective β isolines at their respective centers of mass. More specifically, the germanium and silicon pairs have, in the average, respectively β ≈ 250 and β ≈ 130. Thus, the silicon devices tend to have, in the average, twice as much current gain and smaller V a magnitude values, implying larger R o magnitudes. The average R o values obtained for the germanium NPN and PNP transistors were smaller than those obtained for the two silicon groups. PNP devices tended to present larger parameter variability for both germanium and silicon, but germanium transistors were -surprisingly -characterized by smaller absolute parameter variations than those observed for silicon counterparts.

Figure 7 depicts the four transistor groups in the more traditional parameter space defined by β and R o , also including the respective unit Mahalanobis ellipses. Unfortunately, the relative properties of the considered transistor groups can hardly be discerned in this space, as many ellipses overlap strongly. In addition, the relative features of the transistor groups, so evident in Figure 6, can hardly be inferred from this parametric representation. This striking contrast of results can be understood as corroborating the efficacy and naturality of the Early space with respect to the more traditional parametrization induced by β and R o . The latter space, however, is underlain by more intuitive electronic interpretation, especially when analyzing/designing circuits. So, it may be interesting to consider the twin representation of transistors in the Early as well as in more traditional parametric spaces, the former being more discriminative and inherently compatible with the geometry of transistor operation and the latter being electronically more intuitive.

III. CONCLUDING REMARKS

The advances of electronics since the development of the transistor have been so impressive and continuous that the critical importance that germanium devices played from the 50's to the 60's tends to be overlooked. The present work reported a brief excursion into the origins of modern electronics. More specifically, new old stock germanium junction transistors were characterized by a recently introduced theoretic-experimentalnumerical approach based on the Early effect, as well as on image processing/analysis and pattern recognition concepts and methods. Two types of NPN and one type of PNP new old stock small signal germanium transistors were scanned electronically and the obtained signals were used to derive the respective characterization in the Early, as well as in the more traditional and electronically intuitive ( β , R o ) space.

Remarkable results were obtained with respect to both the main reported investigations, namely the characterization of germanium junction transistors and their comparison with more modern silicon BJT devices. Perhaps the principal contribution of this work is the characterization of the considered four groups of devices -NPN (Ge), PNP (Ge), NPN (Si) and PNP (Si) -in both the Early and the more traditional (β, R o ) spaces. This characterization was performed in terms of the center of mass (average) and dispersion (Mahalanobis distance, derived from the respective covariance matrices), as well as βindexed isolines. A surprisingly well-defined distribution of the transistor types resulted in the Early space, with the NPN devices exhibiting greater parametric variation for both silicon and germanium. Yet and remarkably, unlike it is sometimes believed, the old germanium devices presented substantially smaller parameter variation than the considered modern silicon devices. The four groups of transistors also resulted remarkably separated, with little overlap between them, indicating that, at least for the adopted devices, these groups have well-defined specific characteristics. Germanium devices tended to have smaller V a magnitude (hence smaller R o ), while yielding s values comparable to those of silicon devices, and β was found to be about half of their silicon counterparts.

At the same time, the mapping of the four groups of transistors in the more traditional and electronically intuitive ( β , R o ) space yielded largely overlapping groups to a level that substantially undermines the identification of interesting relationships observed for the Early mapping. It is believe that, because of their complementary features, these two spaces could be used jointly while analyzing and designing devices and circuits.

It should be reminded that the reported results are specific to the devices, methods and configurations adopted in this and previous works [START_REF] Da | Characterizing complementary bipolar junction transistors by Early modelling, image analysis, and pattern recognition[END_REF]. In particular, a limited number of new old stock alloy junction transistors were used, so that distinct results can be eventually obtained for other technologies (e.g. grown junction) and even for other models of alloy junction devices. Additional research is needed to complement the characterization under other circumstances and for additional device types. In addition, it is important to bear in mind that the reported results were derived from new old stock devices from the 50's and 60's, so that it becomes difficult to identify if the obtained characteristics derive from germanium material itself or from the fabrication methods and technology used in the 50's and 60's. It remains an interesting question to verify how modern junction germanium transistors would compare to the here obtained results for new old stock. It would also be worth investigating further the influences of the features observed for the considered devices in typical circuit applications, including the determination of gains, input and output resistances, distortion, and other properties. Of particular interest would be to perform AC analysis of the devices and circuits, as the present work was limited to DC estimation of the transistor features, as it is known that reactive components of transistors can strongly influence AC circuit operation. Yet, the features obtained for the considered new old stock germanium transistors are interesting themselves, especially in the sense that they tend to populate areas of the Early space not well covered by silicon devices, therefore providing valuable design alternatives regarding distinct V a and s features. As a matter of fact, it would be of great potential value to have prototypes of additional devices (such as optocoupler, high frequency and high power transistors, MOSFET, silicon-germanium heterojunction, Darlington, etc.) incorporated into the obtained Early space, so as to achieve a kind of general atlas that could be used to assist electronic analysis and design. A possible reason why such maps have not become widely available is the fact that the parametric mapping of devices into more traditional spaces, as hinted by the here considered and reported cases, tend to imply larger overlap between the groups. The neat separation obtained in the Early space seems to substantiates this approach as providing a more natural and inherently effective means for mapping and studying junction transistors, as well as other similar devices. Only two regions result without overlap, namely those corresponding to silicon transistors but, even so, the region corresponding to their PNP group is severely constrained by overlaps. It should be observed that the orientation of the ellipses in this figure appear not to be visually aligned with the respective groups is a consequence of the fact that the y-axis values are much larger than those at the x-axis.

FIG. 1 :

 1 FIG. 1: The geometrical set-up of Early modeling, including the chosen region of operation (defined by 0 ≤ I C ≤ C,max and 0 ≤ V C ≤ V C,max ), as well as the load line defined by the load resistance L and V C,max . The lines radiating from (Va,0) are henceforth called isolines as they are defined and respectively indexed by constant values of I B .

FIG. 3 :

 3 FIG.3:The accumulator array obtained from equispaced version of the isolines for the transistor in Figure2. A well-defined peak is defined by the intersections of those isolines in the accumulator array. The Va coordinate of the intersection point is taken as the most likely estimation of the Early voltage for that transistor.

FIG. 4 :FIG. 5 :

 45 FIG. 4: Mapping of the three considered groups of germanium junction transistors in the Early space. The two NPN groups, identified as G1 ad G2, present great overlap, but differ substantially from the PNP group G3. The isoline for β = 130, also represented by the dashed curve in this figure, passes very near the center of mass (average) of the NPN and PNP groups.

FIG. 6 :

 6 FIG. 6: The considered four prototypical groups of devices -NPN (Ge), PNP (Ge), NPN (Si) and PNP (Si) -represented in the Early space. The respective dispersions are reflected in the ellipses, while the means are represented as "+". The isolines corresponding to β = 250 and β = 130 are also given (dotted and dashed lines, respectively). The prototypical (V C , I C ) characteristic surfaces for each group are illustrated by respective separated graphs. The overlapping regions are identified in gray.

TABLE I :

 I Some of the main contrasting physical and electronic properties of germanium and silicon.

Acknowledgments.

Luciano da F. Costa thanks CNPq (grant no. 307333/2013-2) for sponsorship. This work has been supported also by FAPESP grants 11/50761-2 and 2015/22308-2.