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Useful papers
The essence of LP-theory of credit risk

Description of credits . In every bank the credit is described by parameters (signs), every parameter has grades. In practice, the number of signs can be 10 - 20 and number of grades in signs can be 2 - 11. For example, credits of natural persons in one of the banks of Russian Federation are described by follwing parameters (signs) and their grades (table 1).

The sign of a credit's repayment - Y (2 grades). Signs of crdit: Z1 - period of loan (4 grades), Z2 - amount of credit (6), Z3 - purpose of credit (3), Z4 - credit history in bank (3), Z5 - possession of bank's credit card (4), Z6 - housing conditions (3), Z7 - possession of expensive property (3), Z8 - age of borrower (3), Z9 - position (4), Z10 - stability of employment (period of working in mentioned company) (4), Z11 - net profit in place of employment (5), Z12 - number of unemployed members of the borrower's family (3).

Quantitative data (period, amount of credit) are divided into intervals. Numbers of grades correspond to intervals (table 2, signs 1,2,8,10,11). This table of credit's description the Customer can present to Executive for control of data.

Formal tabular presentation of bank's credit statistics. Data about bank's credits can be presented as a table (table 2). Credits are presented in strings i=1,2,...,N. Parameters (signs) Z1,...,Zj, ..., Zn, are presented in columns. Signs have grades Zjr, r=1,2,...,Nj; j=1,2,...,n,. Grades are placed in squares of table. In first column of table 2 there is the parameter of credit's effectiveness Y.

Table 1. Description of signs and grades of credit

Sign's number
Sign's denomination
Grade's number
Grades of sign
1
Period of loan
1
2
3
4
6 months and less
From 6 months to 1,5 year
From 1,5 year to 5 years
From 5 years to 15 years
2
Amount of credit
1
2
3
4
5
6
less than 45 000 roubles
From 45 000 roubles to 100 000 roubles
From 100 000 to 200 000 roubles
From 200 000 roubles to 300 000 roubles
From 300 000 roubles to 500 000 roubles
More than 500 000 roubles
3
Purpose of credit
1
2
3
Express-credit
Consumer
Mortgage
4
Credit history in bank
1
2
3
Conscientious credit history
Acceptable credit history
Did not make loans before
5
Possesion of bank credit cards
1
2
3
4
No card
VISA Electron(Cirrus/Maestro, ICB-card)
VISA Classic(Eurocard/Mastercard Mass)
VISA Gold (Eurocard/Mastercard Gold)
6
Housing conditions
1
2
3
House, flat in property
Live in municipal flat or rent flat
Other conditions
7
Possession of expensive property
1
2
3
There is no such property
Car, manufactured not early than 3 years before crediting
Market securities with prices not less than 100 USD
8
Age of borrower
1
2
3
18 - 25 years
26 - 50 years
50 - 75 years
9
Position
1
2
3
4
High level manager, director of company
Middle level mamager, Head of department
Highly qualified specialist
Specialist
10
Stability of employment (period of working in mentioned company)
1
2
3
4
Less than 2 years
From 2 to 4 years
From 4 to 6 years
More than 6 years
11
Net profit in place of employment
1
2
3
4
5
Less than 10 000 roubles/year
From 10 000 roubles/year to 15 000 roubles/year
From 15 000 roubles/year to 30 000 roubles/year
From 30 000 roubles/year to 50 000 roubles/year
More than 50 000 roubles/year
12
Number of unemployed members of the borrower's family
1
2
3
There are no such members
Less than 2 members
2 members and more

Every credit has parameter (sign) of repayment Y, which have two grades: grade 1 ("good") or grade 0 ("bad"). Finally, there is a table, where credits are in strings, the grade of credit's repayment is in first column, grades for signs of credits are in other columns.

Grades are considered as casual events or events-grades. In general, grades are unordered and it is not known, for example, the grade 3 worse or better than grade 4 for final event. Final event has grades also. Set of credits divided into two classes-grades: class 1 - credit was repaid; class 0 - credit default.

Events-grades for every parameter (sign) form group of incompatible events (GIE). Largest number of possible combinations (various credits) is equal:

Nmax = N1 * N2 ... * Nj * ... * Nn,


where N1, N2, ..., Nj, ..., Nn - numbers of grades in parameters (signs).

A number of credits in bank's statistics have to be not less 20*n, where n - a number of signs for description of credits.

Table 2. Credits and their signs and grades

Number of credit
Sign of credit's effectiveness, Y
Sign 1,
Z1
...
Sign j,
Zj
...
Sign 12,
Zn
1
...
...
2
...
...
...
...
...
...
...
...
i
Yr
...
Zjr
...
...
...
...
...
...
...
...
N
...
...


Impersonal presentation of statistics on bank's credits. Impersonal statistics on bank's credit is presented as a tabular file (table 3). This file without first string of identificators of description of credits is sent to Executive by E_mail.


Table 3. Fragment of the file with impersonal statistics of bank's credits

1000 {number of credits}
20 {number of signs}
4
10
5
11
10
5
5
4
4
3
4
4
5
3
3
4
4
2
2
2
{number of grades in every sign}
1
1
3
5
3
3
1
2
4
2
1
4
2
1
3
1
1
3
1
1
1
1
1
2
5
1
5
1
3
2
3
1
2
1
2
3
1
2
3
2
1
1
1
2
2
3
10
2
2
4
2
2
1
4
1
1
3
1
1
2
1
1
1
1
1
2
5
1
4
1
3
3
3
1
2
1
2
3
1
2
2
2
1
2
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1
1
1
5
1
3
1
2
2
3
1
4
1
3
3
2
1
2
2
1
2
1
4
3
5
4
3
1
1
4
2
1
4
3
5
3
2
2
1
1
1
1
0
2
4
3
4
5
3
1
1
2
1
4
4
1
3
1
1
1
1
1
1


General number of credits, number of signs and number of grades in every sign are given by Customer.

Numbers of good and bad credits are calculated in file automatically. These data are used for adjustment of program of training of LP risk model. Also, the calculation of number of equally descripted credits in statistical data is performed and grades of signs, which are not used for description of credits, are determined. It permits to control data about bank's credits and results of training of LP risk model.

For credit risk of juridical persons the categories of borrower are considered as grades 1, 2, 3 :. This parameter of credit's description have to be placed in last column of table 3 and we calculate additional risk attributes for this parameter.

Basic equations and objective function of optimization (training). Signs of credit and their grades are considered as casual events: events-signs and events-grades. These events with certain probability lead to credit default.

Credit default scenario is associative and formulate for whole set of possible events: default is occur, if there is any one, there are any two, ... or all initiative events. Note, there is no known scoring technique for credit risk estimation that is using similar scenario.

Logical variable Zj is equal to1 with probability Pj, if sign j led to default, and equal to 0 with probability Qj = 1 - Pj in other case. Logical probability Zjr, corresponding to grade r of sign j, is equal to 1 with probability Pjr and equal to 0 with probability Qjr = 1 - Pjr . Vector Z(i)=(Z1, :, Zj, :, Zn) describes object i from table 2. For description of object i variables Zjr for grades of signs of this object i are placed instead of logical variables Z1 , :, Zj , :, Zn.

Logical function (L-function) of risk of credit default

L-function of risk of credit default after its orthogonalization:

Probabilistic model (P-model) of risk of credit default:

"Arithmetics" in P-model of risk is: for final event the value of risk is within interval [0, 1] under any probabilities of initiating events.

Objective function for optimization (identification) of P-model of risk of credit default on statistical data of bank is formulated so: a number of correctly classified credits ( Fcor ) have to be maximal


                                 Pjr

where Ngg, Nbb - are numbers of credits, classified as "good" and "bad" by statistics,
and P-model (coincident estimations).

Transparency of LP-model of credit risk and results of estimation and analysis of risk is provided with possibility to calculate contributions of signs and grades in credit risk, in average risk of all set of bank's credits and in accuracy (objective function) of LP-model of credit risk. These contributions can be presented aditionally according to agreement with bank.


The form of presentation of results
1. In result of training (identification) on bank's statistics, following attributes of trained LP-model of risk are calculated ÒÔÒÛÓÕÔÙ ÀÓÕÞÈÜÜÀÝ ùô-ØÀÆÈÙÛ ÒÛ£ßÒ:


N
 - number of credits in bank's statistics;
Ng
 - number of "good" credits in statistics;
Nb
 - number of "bad" credits in statistics;
Pm
 - average risk of bank's credits;
Pad
 - admissible credit risk;
Pmin
 - minimal credit risk in statistical data;
Pmax
 - maximal credit risk in statistical data;
Em
 - average mistake in classification of credits;
Eb
 - mistake in classification of "bad" credits;
Eg
 - mistake in classification of "good" credits;
Egb
 - asymmetry ratio in classification of "good" and "bad" credits (ratio of numbers of wrongly classified "good" and "bad" credits);
Fmax
 - number of correctly classified credits in bank's statistics;
Feff
 - effectiveness of technique - reduction of number of wrongly classified credits by LP-technique in comparison with technique, used by bank;
F1
 - reduction of number of wrongly classified credits due to removal sign 1;
F2
 - reduction of number of wrongly classified credits due to removal sign 2;
...
Fn
 - reduction of number of wrongly classified credits due to removal sign n.


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