Combination of linear discriminant analysis and expert opinion for the construction of credit rating models: The case of SMEs

Figures & data

Table 1. The list of quantitative variables

Class	The quantitative variables ($V_j$,$1\le j\le 16$)
$C_1:Activity$	$V_1$ = Sales/turnover (ST) $V_2=$ Number of employees $V_3=$ Profit growth $V_4=$ Age of the company
$C_2:Returnofinvestment$	$V_5=\frac{NetProfit}{Equity}$$V_6=\frac{NetProfit}{Sales/turnover\left(ST\right)}$
$C_3:Solvability$	$V_7=\frac{Financialexpenses}{Sales/turnover\left(ST\right)}$$V_8=\frac{DLMT\left(Debtlongandmeduimterm\right)}{Equity}$$V_9=\frac{NetDebt}{Equity}$
$C_4:Liquidity$	$V_{10}=\frac{\left(\mathrm{A}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}+\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{s}.\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{h}\right)}{\left(\mathrm{L}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}.\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{h}+\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}-\mathrm{A}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}-\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{y}\right)}$
$C_5:financialstructure$	$V_{11}=\frac{WorkingCapital}{CurrentAssets}$$V_{12}=\frac{Equity}{TotalAssets}$
$C_6:Turnover$	$V_{13}=\frac{Sales/turnover\left(ST\right)}{Netfixedassets+WCR}$ where WCR is $workingcapitalrequirements$$V_{14}=\frac{inventory}{Sales/turnover\left(ST\right)}$$V_{15}=\frac{AccountsReceivable}{Sales/turnove\left(ST\right)}$$V_{16}=\frac{AccountsPayable}{Totalpurchase}$

Table 2. The list of qualitative variables

Theme	The qualitative variables ($q_m$)
$T_1$: The sector of activity	$(q_1=q_{1T1}$): Sector fault rate $\left(q_2=q_{2T1}\right)$: Regulatory impact next year $\left(q_3=q_{3T1}\right)$: Exposure to natural risk
$T_2$: The company’s positioning and competition	$\left(q_4=q_{1T2}\right)$: Competitive position and intensity $\left(q_5=q_{2T2}\right)$: Barriers and new entrants $\left(q_6=q_{3T2}\right)$: International Competition
$T_3$: The concentration and position of the counterparty vis-à-vis its suppliers and customers	$\left(q_7=q_{1T3}\right)$: Customer concentration $\left(q_8=q_{2T3}\right)$: Supplier concentration $\left(q_9=q_{3T3}\right)$: Positions vis-à-vis suppliers and customers
$T_4$: Quality and structure of management	$\left(q_{10}=q_{1T4}\right)$: Succession planning and business continuity $\left(q_{11}=q_{2T4}\right)$: Experience Chairman $\left(q_{12}=q_{3T4}\right)$: Seniority of the principal operational staff $\left(q_{13}=q_{4T4}\right)$: Capital distribution $\left(q_{14}=q_{5T4}\right)$: Compliance with the accounting documents delivery schedule $\left(q_{15}=q_{6T4}\right)$: Performance last crisis $\left(q_{16}=q_{7T4}\right)$: Existence of insurance for corporate officers.
$T_5$: The company’s history with banking	$\left(q_{17}=q_{1T5}\right)$: Number of payment incidents in the last 12 months $\left(q_{18}=q_{2T5}\right)$: Percentages of unpaid bills over the last 12 months
$T_6$: Relations with banks	$\left(q_{19}=q_{1T6}\right)$: Number of banks related to the company

Table 3. Univariate analysis and choice of discriminant variables

hypothesis	Test statistic	Decision
$\left\{\begin{array}{c}{H}_0:equalityofgroupmeans\\ H_1:inequalityofgroupmeans\end{array}\right.$	Fisher’s Ratio	p-value of the Fisher distribution is inferior to 5%, $H_0$ is rejected and the variable is retained

Table 4. Wilks’ lambda

hypothesis	Test statistic	Decision
$\left\{\begin{array}{c}{H}_0:equalityofgroupmeans\\ H_1:inequalityofgroupmeans\end{array}\right.$	Wilks’ lambda ($\mathrm{\Lambda }$)	p-value of the Lamda distribution^2, is inferior to 5%, $H_0$ is rejected

Table 5. The Confusion matrix

			Predicted classification		Total
			0	1
Actual classification	$n=n_1+n_2$	0	$n_{10}$	$n_{11}$	$n_1=n_{00}+n_{01}$
		1	$n_{20}$	$n_{21}$	$n_2=n_{20}+n_{21}$
	%	0	$n_{10}/n_1$	$n_{11}/n_1$	100%
		1	$n_{20}/n_2$	$n_{21}/n_2$	100%

Table 6. Functions at group centroids

		discriminating function for $\mathit{Y}=1$
group	number	The mean of the scores
0	$n_1$	$\alpha$
1	$n_2$	$\beta$

Table 7. The classification table

	Healthy ($\mathit{Y}=1$)	Default ($\mathit{Y}=0$)
Healthy ($Y=1$)	True Healthy ($TH$)	False Healthy ($FH$)
Default ($Y=0$)	False Default ($FD$)	True Default (TD)

Table 8. The rating grid

Class	$\mathit{A}$	$\mathit{B}$	$\mathit{C}$	$\mathit{D}$	$\mathit{E}$	$\mathit{F}$	$\mathit{G}$	$\mathit{H}$
Rating score	[86–100]	[76–86[	[65–76[	[55–65[	[46–55[	[40–46[	[30–40[	<30
Qualification	Excellent	Very good	Good	Quite good	Average	Passable	Mediocre	Very Mediocre

Table 9. The probability of default by class

Class	A	B	C	D	E	F	G	H
Rating score	[86–100]	[76–86[	[65–76[	[55–65[	[46–55[	[40–46[	[30–40[	<30
Number$\left\{Y=0\right\}$	$n_{A0}$	$n_{B0}$	$n_{C0}$	$n_{D0}$	$n_{E0}$	$n_{F0}$	$n_{G0}$	$n_{H0}$
Number $\left\{Y=1\right\}$	$n_{A1}$	$n_{B1}$	$n_{C1}$	$n_{D1}$	$n_{E1}$	$n_{F1}$	$n_{G1}$	$n_{H1}$
Total number	$n_A$	$n_B$	$n_C$	$n_D$	$n_E$	$n_F$	$n_G$	$n_H$
Probability of Default ($PD$)	$n_{A0}/n_A$	$n_{B0}/n_B$	$n_{C0}/n_C$	$n_{D0}/n_D$	$n_{E0}/n_E$	$n_{F0}/n_F$	$n_{G0}/n_G$	$n_{H0}/n_H$

Table 10. Rating of the scoring criteria for credit portfolio managers

Qualifications	low	medium	high
ratings	1	2	3

Table 11. Weighting of credit portfolio managers as a function of the score

Score	$\left(\mathbf{4}\mathbf{-}\mathbf{6}\right)$	$\left(\mathbf{7}\mathbf{-}\mathbf{8}\right)$	$\left(\mathbf{9}\mathbf{-}\mathbf{10}\right)$	$\left(\mathbf{11}\mathbf{-}\mathbf{12}\right)$
weighting	10%	25%	50%	75%

Table 12. The portfolio structure in terms of default

	Number of enterprises	Percentage of enterprises
Healthy enterprises	1333	92,12%
Enterprises in default	114	7,88%
Total	1447	100,00%

Table 13.. ($VC{B}_0$) and ($V_{efi}$) by rating class

	A	B	C	D	E	F	G	H	total
$V_{efi}$	336,40	596,93	3 129,18	2 354,36	1 453,32	615,08	179,16	223,88	8 888,30
$VC{B}_0$	170,06	456,10	2 539,04	1 944,17	1 404,08	538,67	150,71	293,54	7 496,37

Table 14. Non-discriminatory variables

Variable	Lambda	F	$\mathit{D}DL$1	$\mathit{D}DL$2	p-value
$V_4$	0,9996	0,5646	1	1445	0,4525
$V_6$	0,9982	2,528	1	1445	0,1120
$V_{11}$	0,9995	0,6139	1	1445	0,4334
$q_2$	0,9999	0,0651	1	1445	0,7985
$q_3$	0,9998	0,1865	1	1445	0,6658
$q_4$	0,9994	0,7704	1	1445	0,3802
$q_5$	0,9999	0,0615	1	1445	0,8040
$q_7$	0,9995	0,5998	1	1445	0,4387
$q_9$	0,9996	0,5625	1	1445	0,4533
$q_{10}$	0,9998	0,2129	1	1445	0,6445
$q_{11}$	0,9996	0,5253	1	1445	0,4686
$q_{13}$	0,9996	0,5253	1	1445	0,4686
$q_{14}$	0,9975	3,5926	1	1445	0,0582
$q_{15}$	0,9999	0,0838	1	1445	0,7721
$q_{16}$	0,9987	1,8025	1	1445	0,1796
$q_{18}$	0,9995	0,6071	1	1445	0,4360

Table 15. The quantitative and qualitative discriminating variables

Variable	Lambda	F	$DDL$1	$DDL$2	p-value
$V_1$	0,9971	4,1184	1	1445	0,0426
$V_2$	0,9937	9,0833	1	1445	0,0026
$V_3$	0,9966	4,8286	1	1445	0,0281
$V_5$	0,9934	9,5609	1	1445	0,0020
$V_7$	0,982	26,3792	1	1445	< 0,0001
$V_8$	0,9925	10,8249	1	1445	0,0010
$V_9$	0,9829	25,1194	1	1445	< 0,0001
$V_{10}$	0,9956	6,2559	1	1445	0,0124
$V_{12}$	0,9954	6,6673	1	1445	0,0099
$V_{13}$	0,9927	10,5841	1	1445	0,0011
$V_{14}$	0,9969	4,3537	1	1445	0,0371
$V_{15}$	0,9965	4,9657	1	1445	0,0260
$V_{16}$	0,9972	3,9544	1	1445	0,0469
$q_1$	0,9948	7,5248	1	1445	0,0061
$q_6$	0,9975	3,8828	1	1445	0,0490
$q_8$	0,9974	3,8818	1	1445	0,0490
$q_{12}$	0,9973	3,8981	1	1445	0,0485
$q_{17}$	0,8414	272,2333	1	1445	< 0,0001
$q_{19}$	0,9919	11,6537	1	1445	0,0006

Table 16. Analysis of the correlation of quantitative discriminant variables

	$V_1$	$V_2$	$V_3$	$V_5$	$V_7$	$V_8$	$V_9$	$V_{10}$	$V_{12}$	$V_{13}$	$V_{14}$	$V_{15}$	$V_{16}$
$V_1$	1,000	0,155	0,004	0,005	−0,084	−0,053	−0,135	−0,133	−0,047	−0,018	0,021	−0,066	0,014
$V_2$	0,155	1,000	−0,054	−0,018	−0,169	−0,064	−0,103	0,009	0,023	−0,231	−0,032	0,134	0,033
$V_3$	0,004	−0,054	1,000	0,128	0,076	0,004	0,060	0,067	0,109	0,008	0,017	−0,009	−0,065
$V_5$	0,005	−0,018	0,128	1,000	−0,021	0,048	0,069	0,087	0,213	−0,084	0,043	0,014	−0,054
$V_7$	−0,084	−0,169	0,076	−0,021	1,000	0,217	0,415	0,195	0,085	0,440	0,009	−0,131	−0,130
$V_8$	−0,053	−0,064	0,004	0,048	0,217	1,000	0,603	0,165	0,259	0,223	0,040	0,070	−0,020
$V_9$	−0,135	−0,103	0,060	0,069	0,415	0,603	1,000	0,297	0,357	0,358	0,033	−0,065	−0,035
$V_{10}$	−0,133	0,009	0,067	0,087	0,195	0,165	0,297	1,000	0,371	−0,027	−0,083	0,079	−0,316
$V_{12}$	−0,047	0,023	0,109	0,213	0,085	0,259	0,357	0,371	1,000	−0,159	0,035	−0,119	−0,339
$V_{13}$	−0,018	−0,231	0,008	−0,084	0,440	0,223	0,358	−0,027	−0,159	1,000	0,055	−0,253	−0,016
$V_{14}$	0,021	−0,032	0,017	0,043	0,009	0,040	0,033	−0,083	0,035	0,055	1,000	−0,006	0,007
$V_{15}$	−0,066	0,134	−0,009	0,014	−0,131	0,070	−0,065	0,079	−0,119	−0,253	−0,006	1,000	0,448
$V_{16}$	0,014	0,033	−0,065	−0,054	−0,130	−0,020	−0,035	−0,316	−0,339	−0,016	0,007	0,448	1,000

Table 17. Results of the Box’s M test

−2Log(M)	Khi^2 (Observed value)	Khi^2 (Critical value)	$DDL$	p-value	$\alpha$
490,1757	463,3248	202,5125	171	< 0,0001	0,05

Table 18. Wilks’ lambda

Lambda	F (observed value)	F (critical value)	$DDL$1	$DDL$2	p-value	alpha
0,8030	19,3640	1,6110	18	1428	< 0,0001	0,05

Table 19. The Confusion matrix

			Predicted classification		Total
			0	1
Actual classification	$n=n_1+n_2$	0	51	63	114
		1	28	1305	1333
	%	0	44,70	55,30	100,00
		1	2,10	97,90	100,00

Table 20. The canonical correlation of variables

Function	eigenvalue	Pct.	Cum.	Canon Cor.
1	0,2440	100,00	100,00	0,4430

Table 21. The functions at group centroids

group	discriminating function for $Y=1$
	The mean of the scores
0	−1,6880
1	0,1440

Table 22. Rating model based on linear discriminant analysis

Rating class	Grid	$PTF$ Healthy	$PTF$ in default	$PD$
A	[86–100]	9	0	0,03%
B	[76–86[	122	2	1,61%
C	[65–76[	450	20	4,26%
D	[55–65[	425	25	5,56%
E	[46–55[	243	20	7,60%
F	[40–46[	66	13	16,46%
G	[30–40[	14	17	54,84%
H	<30	4	17	80,95%
		1333	114

Table 23. Explicit estimation of the probability of default by class according to experts

Rating class	Number of default according to size of $\mathit{P}TF$			Probability of default by rating class according to size of $\mathit{P}TF$			Mean $\mathit{P}{D}_{e,K}$
	100	1 000	10 000	$\mathit{P}{D}_{100}$	$\mathit{P}{D}_{1000}$	$\mathit{P}{D}_{10000}$
A	0	0	6	0,00%	0,00%	0,06%	0,02%
B	0	2	12	0,00%	0,20%	0,12%	0,11%
C	0	30	35	0,00%	3,00%	0,35%	1,12%
D	1	9	90	1,00%	0,90%	0,90%	0,93%
E	3	25	300	3,00%	2,50%	3,00%	2,83%
F	8	100	1 000	8,00%	10,00%	10,00%	9,33%
G	21	200	2 000	21,00%	20,00%	20,00%	20,33%
H	55	550	5 500	55,00%	55,00%	55,00%	55,00%

Table 24. Estimate of expected losses by class according to experts

Rating class	The expected loss ($EL$) according the $\mathit{E}AD$			The implicit $\mathit{P}D$			Mean $\mathit{P}{D}_{e,K}$
	100 000	1 000 000	10 000 000	$\mathit{P}{D}_1$	$\mathit{P}{D}_2$	$P{D}_3$
A	0	0	5000	0,00%	0,00%	0,11%	0,04%
B	0	1 000	7000	0,00%	0,22%	0,16%	0,13%
C	500	5 000	25 000	1,11%	1,11%	0,56%	0,93%
D	1 000	7 500	100 000	2,22%	1,67%	2,22%	2,04%
E	2 200	12 000	140 000	4,89%	2,67%	3,11%	3,56%
F	4 500	22 500	100 000	10,00%	5,00%	2,22%	5,74%
G	12 000	50 000	2 000 000	26,67%	11,11%	44,44%	27,41%
H	25 000	200 000	3 000 000	55,56%	44,44%	66,67%	55,56%

Table 25. The probability of default of the experts retained for the modelling

Rating class	A	B	C	D	E	F	G	H
$P{D}_{e,K}$	0,03%	0,11%	1,01%	1,47%	3,18%	7,52%	23,85%	55,28%

Table 26. Bayesian rating models

Rating class	A	B	C	D	E	F	G	H
$P{D}_K$ $LDA$ model	0,03%	1,61%	4,26%	5,56%	7,60%	16,46%	54,84%	80,95%
$P{D}_{e,K}$	0,03%	0,11%	1,01%	1,47%	3,18%	7,52%	23,85%	55,28%
Model 1 $P{D}_{a,K}$($\epsilon$=25%)	0,03%	1,24%	3,45%	4,54%	6,50%	14,23%	47,09%	74,53%
Model 2 $P{D}_{a,K}$($\epsilon$=50%)	0,03%	0,86%	2,64%	3,52%	5,39%	11,99%	39,35%	68,12%

Table 27. The unexpected loss according to the $LDA$ model

Rating class	$\mathit{P}{D}_c$	${\sum }_{{\mathit{i}\mathbf{=}\mathbf{1}}^{n_{ic}}\left(N\left(\frac{G\left(P{D}_c\right)+\sqrt{R_i}G\left(0,999\right)}{\sqrt{1-R_i}}\right)-P{D}_c\right)\ast EA{D}_i}$	${\beta }_c$	$U{L}_0$
A	0,03%	2,47	85,76%	2,12
B	1,61%	83,87	54,76%	45,93
C	4,26%	288,11	51,57%	148,58
D	5,56%	610,83	50,84%	310,57
E	7,60%	478,68	50,06%	239,65
F	16,46%	266,12	48,43%	128,89
G	54,84%	48,05	46,60%	22,39
H	80,95%	65,59	46,17%	30,29
				928,40

Table 28. The Bayesian unexpected loss

Class	$\mathit{P}{D}_{c1}$$(\epsilon$=25%)	$\mathit{P}{D}_{c1}$$(\epsilon$=50%)	${\sum }_{i=1}^{n_{ic}}\left(N\left(\frac{G\left(P{D}_c\right)+\sqrt{R_i}G\left(0,999\right)}{\sqrt{1-R_i}}\right)-P{D}_c\right)\ast EA{D}_i$		$\mathit{U}{L}_1$	$\mathit{U}{L}_2$
	Model 1	Model 2	Model 1	Model 2
A	0,03%	0,03%	2,28	2,05	1,96	1,76
B	1,24%	0,86%	77,77	70,82	43,38	40,62
C	3,45%	2,64%	267,52	244,67	139,60	129,73
D	4,54%	3,52%	568,33	521,57	292,06	271,86
E	6,50%	5,39%	446,12	410,64	225,04	209,12
F	14,23%	11,99%	256,87	244,68	125,12	120,02
G	47,09%	39,35%	52,65	56,15	24,64	26,41
H	74,53%	68,12%	71,11	75,24	32,89	34,87
					884,70	834,41

Figure 1. Comparison of the performance of correlated variables $V_8$ and $V_9$.

Figure 2. Performance of the linear discriminant analysis model.

Figure 3. The distribution of healthy and defaulting enterprises and the probability of default by rating class.

Table A1. Explanation and significance of quantitative variables

Class	The quantitative variables ($V_j$,$1\le j\le 16$)
$C_1:Activity$	This class determines the impact of the size of the company on the probability of default. Indeed, the size is determined by the turnover, the number of employees, Profit growth and the age of the company;
$V_1$	$V_1$ = Sales or Turnover (ST)
	This variable represents total sales, since a significant turnover is synonymous with the existence of liquidity to honour commitments. As a result, it must be inversely proportional to the probability of default.
$V_2$	$V_2=$ Number of employees
	The size of the company and its stability can be measured by the number of employees
$V_3$	$V_3=$ Profit growth
	Profit growth or decline may be a sign of an upgrading or deterioration of the company’s risk profile
$V_4$	$V_4=$ Age of the company
	The age of the company may explain the failure because new companies are more exposed to failure than older companies. Indeed, in general, the old companies are in a balanced position with regard to the market, to the customers and competitors.
$C_2:Returnofinvestment$	The return on investment is determined by the ratio of net profit to equity and turnover. indeed, low profitability can be synonymous with survival risk and high profitability can be synonymous with a high-risk investment such as high-risk sectors
$V_5$	$V_5=\frac{NetProfit}{Equity}$
	This ratio measures the return on equity. Indeed, low return can mean that the company is having difficulty establishing itself in the market and high return can mean that the company is taking unmeasured risks
$V_6$	$V_6=\frac{NetProfit}{Sales\mathrm{o}\mathrm{r}Turnover\left(ST\right)}$
	This ratio measures the proportion of net profit in turnover. Indeed, a low ratio may mean that the company practices low margins dictated by market conditions or has difficulties mastering the costs of production and marketing. Whereas a high ratio means that sales turnover is high or that the company is dominant in its market so that it can impose these prices by mastering costs.
$C_3:Solvency$	Solvency is measured by the proportion of financial expense in turnover as well as the proportion of debt to equity in the company’s financing. Indeed, high financial costs can be a sign of the company’s difficulty and a high proportion of the debt can mean that the company is over-indebted.
$V_7$	$V_7=\frac{Financialexpenses}{Sales\mathrm{o}\mathrm{r}\mathrm{T}urnover\left(ST\right)}$
	This ratio measures the proportion of financial charges in turnover. a high ratio may mean that the company regularly uses expensive banking services due to long customer lead times payment, lack of liquidity or difficulties in debt collection.
$V_8$	$V_8=\frac{DLMT\left(Debtlongandmeduimterm\right)}{Equity}$
	This ratio measures the proportion of debt log and medium term in the company’s financing. Indeed, a high level of credit financing can mean that the company is over-indebted or small capitalized.
$V_9$	$V_9=\frac{NetDebt}{Equity}$
	Net debt includes $Debtlongandmeduimterm$(DLMT) and short-term debt (DST). As a result, it is an extension of the $V_8$ ratio. in fact, it measures the company’s dependence on debt both in the short term and in the medium and long term
$C_4:Liquidity$	Liquidity represents the ability of current assets to cover current liabilities. Indeed, default depends on the company’s ability to honor these short-term commitments.
$V_{10}$	Let be: A =$\mathrm{A}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$; B =$\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{s}.\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{h}$; C =$\mathrm{L}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}.\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{h}$; D =$\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$; F =$\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{y}$:
	$V_{10}=\frac{\left(\mathrm{A}+\mathrm{B}\right)}{\left(\mathrm{C}+\mathrm{D}-\mathrm{A}-\mathrm{F}\right)}$
	This ratio represents the rate of coverage of current liabilities and cash liabilities by accounts receivable and cash assets. Indeed, a coverage in excess of 100% is a sign of the company’s solvency.
$C_5:financialstructure$	The financial structure represents the balance between long-term assets and long-term liabilities as well as the proportion of total assets financed by equity.
$V_{11}$	$V_{11}=\frac{WorkingCapital}{CurrentAssets}$
	This ratio represents the proportion of current assets financed by excess of permanent capital over fixed assets. indeed, the company can be in a comfortable position if the coverage rate is significant without going so far as to have an inactive cash position.
$V_{12}$	$V_{12}=\frac{Equity}{TotalAssets}$
	This ratio represents the proportion of equity in the financing of the company’s total assets. Indeed, a low ratio is synonymous with undercapitalization
$C_6:Turnover$	This class is dedicated to the study of the rotation of certain aggregates such as enterprise value, inventory, Accounts receivable and Accounts payable. Indeed, a high rotation is significant for business dynamism and business continuity.
$V_{13}$	$V_{13}=\frac{Sales\mathrm{o}\mathrm{r}\mathrm{T}urnover\left(ST\right)}{Netfixedassets+WCR}$ where WCR is working capital requirement
	The sum of Net fixed assets and WCR of represents the economic value of the company. As a result, this ratio represents the number of times per year that turnover can repay the economic value of the company. Indeed, it means that the company’s activity can repay capital, medium and long-term debts and the working capital requirement a number of times per year equivalent to the value of the ratio.
$V_{14}$	$V_{14}=\frac{inventory}{Sales\mathrm{o}\mathrm{r}Turnover\left(ST\right)}$
	This ratio determines the velocity of stock rotation. Indeed, a high rotation can be synonymous with a dynamic company on its market
$V_{15}$	$V_{15}=\frac{AccountsReceivable}{Sales\mathrm{o}\mathrm{r}Turnove\left(ST\right)}$
	This ratio makes it possible to assess the payment terms granted to customers and the effectiveness of the collection policy. Indeed, a low ratio shows that the company practices short lead times, which can be synonymous with a strong market position and recovery efficiency.
$V_{16}$	$V_{16}=\frac{AccountsPayable}{Totalpurchase}$
	This ratio measures the turnover velocity of supplier credit in relation to total purchases. Indeed, it measures the company’s ability to negotiate payment terms with these suppliers. Supplier credit is measured in number of days of purchases. A high ratio can be synonymous with a solid position of the company with regard to these suppliers.

Table A2. Explanation and significance of qualitative variables

Theme	The qualitative variables (${\mathit{q}}_m$,$1\text{lem}\le 19$)
$T_1$: The sector of activity	This theme contains factors external to the company that may correlate with the company’s failure, such as industry, regulation and natural risk.
$q_1$	$q_1=\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{a}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}$
	This variable can explain the company’s default. Indeed, companies in sectors with a high failure rate may have a higher probability of default than companies in sectors with a low failure rate.
$q_2$	$q_2=$ Regulatory impact next year
	Changes in regulations, such as import or export restrictions, increases in taxes, payment delays and labour law, may have a negative impact on the company’s business. consequently, the assessment of the impact of the regulations on the company may influence the default prediction
$q_3$	$q_3=$ Exposure to natural risk
	Climatic conditions can affect the good functioning of certain companies, particularly those in the agri-food sector. As a result, exposure to natural risks such as flooding and drought can have an impact on the probability of default
$T_2$: The company’s positioning and competition	This theme concerns the competitive environment of the company and the difficulties that the company may encounter when setting up. Indeed, the company’s position on the market as well as barriers to entry and national and international competition can have an impact on the company’s failure.
$q_4$	$q_4=\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}$
	The company’s position may be monopoly, market leader or among the top 5, top 10 or other. The position may be synonymous with solidity or fragility as a result, it may in some cases explain the default.
$q_5$	$q_5=$ Barriers and new entrants
	The company may be an old or new entrant company and the sector may be open access or requires entry conditions that represent barriers for the company such as technical expertise, a given level of capital, regulatory authorizations etc. The existence or absence of barriers may explain the defaults in certain cases.
$q_6$	$q_6=$ International Competition
	Competition can be national or international. Indeed, for SMEs, international competition can be a major constraint. Consequently, the existence of international competition can encourage default and increase the probability of default.
$T_3$: The concentration and position of the counterparty vis-à-vis its suppliers and customers	The company’s relationship and position with its customers and suppliers can be decisive for its survival and business continuity. Indeed, the company can be a maker or a taker of prices and deadlines with its customers and suppliers, which affects its liquidity. It can also be dependent or independent of its customers and suppliers. As a result, this theme can influence the probability of default.
$q_7$	$q_7=\mathrm{C}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$
	Customer consultation can be crucial for the survival of the company. Indeed, the company may have a single customer, a small number of customers or a diversification of customers. In the first case, the company is dependent on its customer because it can cease its activity following the failure of its customer, the non-conformity of these products and services with the requirements of its customer or following a conflict with him. As a result, the client’s concentration may influence the probability of default.
$q_8$	$q_8=$ Supplier concentration
	The concentration of suppliers can also be decisive for the survival of the company. Indeed, the company may be in a position of strength or weakness vis-à-vis these suppliers because it may suffer if there is only one supplier from the latter’s abuses in terms of price-fixing and payment terms. As a result, supplier concertation can influence the probability of default.
$q_9$	$q_9=$ Positions vis-à-vis suppliers and customers
	The company can be a maker or taker of prices and payment terms with its customers and suppliers, which allows it to define a set of situations defined by the couple (position with regard to the customer, position with regard to the supplier). Each situation can have an impact on the liquidity and cost of products and services. As a result, this variable can influence the probability of default.
$T_4$: Quality and structure of management	This theme integrates all the variables relating to the structure and quality of internal management adopted by the company. Indeed, it covers the capital structure, the experience of the chairmen and senior managers, the successor plan, the timely production of accounting and financial information, historical performance in the event of a crisis and the existence of insurance to cover corporate officers. These variables can influence the probability of default.
$q_{10}$	$q_{10}=\mathrm{S}\mathrm{u}\mathrm{c}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{b}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}$
	The existence of a continuity and succession plan is very important for the survival of the company in the short and long term. Indeed, it ensures that the company is able to ensure the transition in the event of the absence of current managers. The absence of this plan can impact the probability of default.
$q_{11}$	$q_{11}=$ Experience Chairman
	The president’s experience in managing the company, measured in number of years, is an important factor in the company’s survival. Indeed, confirmed experience allows to assure partners about the company’s future, particularly in terms of commercial conflict, debt recovery, banking relations and market mastery. As a result, it can have an impact on the probability of default.
$q_{12}$	$q_{12}=$ Seniority of the principal operational staff
	This variable includes the top managers in the assessment of the probability of default. Indeed, a confirmed experience of the latter can ensure partners on the future of the company, notably in the absence of the president and a succession plan.
$q_{13}$	$q_{13}=\mathrm{C}\mathrm{a}\mathrm{p}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$
	The concentration of capital in a single hand or in a family represents a high risk because the company will be dependent on the person who has a majority share in the capital. Indeed, a division of capital reduces this risk. As a result, the concentration may increase the probability of default
$q_{14}$	$q_{14}=$ Compliance with the accounting documents delivery schedule
	The timely production of accounting and financial information and its submission to the banker can be synonymous with transparency and better monitoring of the company’s financial situation. The delay can increase partners’ uncertainty about the company’s future, which can lead to an increase in the probability of default.
$q_{15}$	$q_{15}=$ Performance last crisis
	Experience in crisis management is an important indicator of the company’s capacity to surmount difficulties, notably, in terms of liquidity and financing. As a result, this variable permits to adjust the probability of default.
$q_{16}$	$q_{16}=$ Existence of insurance for corporate officers
	Existence of insurance for corporate officers to cover the loss caused by these officers in the event of mismanagement. As a result, it can reduce the probability of default
$T_5$: The company’s history with banking	This theme includes non-payment incidents registered by the bank. Indeed, these incidents are signs of the deterioration of the company’s risk profile. The existence of recent incidents may increase the probability of default.
$q_{17}$	$q_{17}=\mathrm{N}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{p}\mathrm{a}\mathrm{y}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}12\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{s}$
	This variable represents the number payment incidents. It shows that the company has liquidity problems
$q_{18}$	$q_{18}=\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{d}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{s}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}12\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{s}$
	Similarly, for this variable, it complements the previous one by including commercial paper emitted by the customer and rejected by the bank. It shows that the company has liquidity problems.
$T_6$: $\mathrm{R}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{s}$	This theme concerns the number of banks in relation to the company. Indeed, for small and medium companies, it is preferable for the relationship to be limited to a single bank; this allows the company to benefit from the bank’s advice and to limit the management charge to a single bank. For big companies with a significant financing requirement, it is preferable for banks to form a consortium to finance it in order to prevent the risk of concentration.
$q_{19}$	$q_{19}=\mathrm{N}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{s}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{n}\mathrm{y}$
	The number of banks in the company must be proportional to its size. The multitude of banks for small and medium-sized structures can be a sign of lack of liquidity and financial mismanagement. As a result, this variable can impact the probability of default