Fuzzy structural risk of default for banks in Southern Africa

Figures & data

Table

Suggestion/Comment by reviewer	Reviewer rating	Revision Made by author/s
Reviewer 2
Title, Abstract and Introduction	Outstanding
Methodology/Materials and Methods	Sound with minor or moderate revisions	The materials and methods used have been improved in terms of sequencing, language editing and overall presentation
Objectives/Hypothesis	Sound
Figures and Tables	Sound
Results/Data Analysis	Sound with minor or moderate revisionsProcedures and techniques used for data collection and analysis are partially adequate.	Sources of research data have been fairly articulated in line with expectations of the reviewer. The author/s took time to read through all paper results and data analysis to align them to the order of the objectives of the study. Presentation of paper results and analysis has been significantly revised to meet comments and suggestions by the reviewer.
Interpretation/Discussion	Sound with minor or moderate revisions Results must be supported by theory or empirical evidence and be well articulated	The interpretation and discussion improvements made to the paper are shown in BOLD in the relevant paper sections in much detail. The paper was then handed over to a language specialist after the above amendments to make sure grammatical issues are smoothened before sending it back to the Journal.
Conclusions	Sound
References	Sound
Compliance with Ethical Standards	Sound
Writing	Sound
Supplementary Information and Data	Sound
Reviewer 3
Title, Abstract and Introduction	Sound
Methodology/Materials and Methods	Sound
Objectives/Hypothesis	Sound
Figures and Tables	Sound
Results/Data Analysis	Sound
Interpretation/Discussion	Sound
Conclusions	Sound
References	Sound
Compliance with Ethical Standards	Sound
Writing	Sound with minor or moderate revisions	The paper writing style has been significantly improved. Paper has been proofread by a language editor to enable it meet the expected writing standard
Supplementary Information and Data	Sound

Figure 1. Assets of a bank sensitised by those of the borrower.

Source: Authors.

Table 1. The architecture of fuzzy systems by Ross et al. (2010)

Structure of Fuzzy Systems (Source: Ross et al., 2010).

Table 2. The process of fuzzification and de-fuzzification of model variables

Variable/Approach	The system applied
Input fuzzy sets	Gaussian
Output fuzzy sets	Triangle
Implication method	AND/OR
Aggregation method	Product (AND)
De-fuzzification method	Centre of gravity

Table 3. The three stages of the study procedure or variables

A. Definitions of variables	B. Definitions of rules	C. Definitions of inferential machine settings
Sets of definitions	Operational research	Aggregations
Membership functions	Database formations	De-fuzzification

Source: Authors.

Table 4. Distribution of banks by their annual structural (S) and Fuzzy KMV (F) DP values and hazard ratios (H) for the period 2008–2020

Year	PD value	08	09	10	11	12	13	14	15	16	17	18	19	20
Bank A	S	16	3.7	3.9	3.0	0.0	1.6	3.2	2.7	2.2	2.4	1.9	2.1	2.0
	F	8.0	2.6	0.3	0.0	0.0	0.1	1.2	1.6	1.0	1.2	1.3	1.6	1.4
	H	2.2	1.8	1.4	0.8	0.9	1.4	1.8	0.8	1.5	1.8	1.6	1.9	1.7
Bank B	S	24	2.7	2.4	2.7	2.9	2.8	2.2	1.9	1.6	1.8	2.1	2.5	1.8
	F	2.0	3.8	0.0	0.0	0.1	2.4	1.8	1.4	1.3	1.5	1.6	2.2	1.5
	H	1.2	1.6	1.5	1.2	1.0	0.8	1.2	0.7	1.4	0.9	1.0	0.5	0.7
Bank C	S	2.8	4.0	3.5	1.5	3.4	3.9	2.8	2.4	2.7	2.3	2.5	1.8	1.6
	F	0.0	0.0	0.0	0.0	0.0	0.0	1.2	1.4	16	1.4	1.5	1.2	1.1
	H	1.2	0.9	1.0	0.8	0.6	1.2	1.8	1.6	1.7	1.8	1.6	1.3	1.5
Bank D	S	0.8	0.4	0.0	0.0	0.0	1.8	1.2	1.6	2.0	1.8	1.4	1.5	1.7
	F	0.4	0.2	0.0	0.0	0.0	0.8	0.6	0.7	1.2	1.0	0.9	0.9	1.0
	H	1.6	0.9	0.7	1.1	0.8	1.6	1.3	1.5	1.6	1.6	1.0	0.8	0.9
Bank E	S	0.5	0.0	0.0	0.0	0.0	0.0	1.2	0.8	1.0	1.6	1.8	1.7	1.5
	F	0.0	0.0	0.0	0.0	0.0	0.0	0.8	0.4	0.4	0.8	0.7	0.9	0.6
	H	1.3	0.8	0.6	0.9	0.7	1.0	1.0	0.9	0.7	1.2	1.4	1.2	1.3
Bank F	S	2.5	4.0	0.0	3.1	3.8	4.9	0.8	2.2	1.5	0.4	0.6	0.3	0.5
	F	1.8	2.1	0.0	0.0	2.2	0.8	0.0	0.7	0.8	0.2	0.2	0.1	0.2
	H	1.8	1.4	1.0	0.8	1.2	1.5	1.3	1.1	0.6	0.4	0.6	0.4	0.5
Bank G	S	42	38	34	30	26	32	34	36	33	48	44	42	40
	F	18	16	12	14	15	17	20	22	24	19	21	23	25
	H	1.4	1.8	2.0	1.7	2.2	1.8	1.6	2.0	1.8	1.4	1.7	0.9	0.8
Bank H	S	56	48	55	47	56	44	54	43	52	48	53	48	56
	F	34	22	24	25	32	38	46	32	36	28	36	29	28
	H	1.3	0.9	1.1	1.6	1.8	1.7	1.4	1.9	1.6	1.5	0.9	0.7	0.6

Table A1. Distribution of banks by their transaction costs (Costs of Ordinary Equity, $K_{es})$ for 2008–2020 (%)

Year	08	09	10	11	12	13	14	15	16	17	18	19	20
A	12.7	12.8	16.1	16.0	12.3	8.7	9.6	10.5	10.2	10.8	11.3	12.7	12.6
B	12.5	13.9	14.4	13.8	14.2	11.6	12.8	13.3	10.6	14.8	12.5	10.9	12.4
C	12.1	10.5	18.0	15.2	16.4	11.9	13.6	15.5	12.3	15.9	11.7	13.6	14.2
D	12.8	13.6	16.8	16.1	15.5	14.3	15.4	12.7	14.1	16.3	13.2	10.6	10.8
E	19.9	13.5	17.8	17.6	16.4	12.4	12.7	11.9	13.8	12.6	13.2	14.5	12.9
F	17.0	11.6	13.4	13.8	14.3	14.1	13.8	12.7	11.5	10.8	12.2	13.7	13.4
G	96.0	12.4	12.8	12.6	10.3	13.8	14.2	14.9	18.6	22.1	24.7	26.3	32.6
H	92.8	14.2	14.6	16.2	13.5	13.7	14.8	15.7	16.8	19.6	23.3	24.6	33.4

Table A2. Banks’ traditional and fuzzy asset standard deviations for 2008–2020 (%)

	08	09	10	11	12	13	14	15	16	17	18	19	20
A-T	16.8	21.6	53.1	0.6	12.6	16.5	11.2	18.3	9.5	16.8	22.7	15.6	36.4
F	18.4	20.8	34.4	10.3	16.2	17.6	13.8	21.6	11.7	18.4	25.2	17.3	28.8
B-T	11.5	4.3	15.8	34.0	23.2	20.0	18.9	21.6	24.2	26.8	19.5	22.7	20.4
F	15.8	12.2	17.9	27.0	21.6	20.0	19.3	17.5	19.6	17.8	20.2	18.4	18.6
C-T	9.8	15.4	8.2	27.4	10.0	24.3	18.6	16.5	20.8	17.2	16.8	13.6	22.4
F	14.9	17.7	14.1	23.7	15.0	22.2	19.8	17.3	17.5	15.4	18.5	16.8	17.4
D-T	8.7	5.2	12.8	7.7	14.8	12.6	10.2	9.4	11.6	9.8	10.7	11.3	12.8
F	14.4	12.6	16.4	13.9	17.4	15.6	13.7	12.9	12.6	11.7	12.9	12.3	15.2
E-T	6.8	8.4	1.7	7.6	8.0	11.2	9.5	8.8	7.6	4.9	6.7	7.8	9.6
F	13.4	14.2	12.5	13.8	14.0	5.60	15.9	15.6	13.8	12.5	14.9	13.7	15.8
F-T	24.6	35.02	33.8	11.0	54.0	32.6	44.4	28.9	48.2	33.7	28.8	44.5	50.8
F	23.0	27.5	26.9	15.5	37.0	27.8	32.6	26.4	36.3	27.1	25.8	32.4	37.6
G-T	72.6	10.5	12.5	14.6	11.2	12.3	14.9	15.7	16.5	18.6	20.4	25.7	27.5
F	68.5	11.6	15.8	16.5	13.3	14.8	15.3	16.7	17.4	19.6	21.9	26.2	28.7
H-T	74.4	12.6	14.3	18.6	8.7	5.4	6.3	9.2	14.8	18.7	20.7	23.5	26.8
F	71.6	14.8	16.8	20.4	8.8	5.6	7.8	10.8	15.1	18.7	22.6	25.5	27.8

Table A3. Distribution of banks by their traditional and fuzzy ROEs for 2008–2020 (%)

Year	08	09	10	11	12	13	14	15	16	17	18	19	20
A-T	13.4	11.0	124.2	42.3	5.6	8.7	10.5	12.8	11.7	13.4	10.8	12.3	14.8
F	−9.6	53.0	258.8	78.0	47.9	56.6	70.5	82.8	75.6	88.4	71.3	74.7	86.6
B-T	16.1	20.1	23.5	20.7	21.9	20.0	18.6	16.4	17.7	18.9	15.8	16.5	17.6
F	7.2	92.3	115.4	83.1	110.7	53.5	86.3	63.6	72.2	68.7	72.8	58.3	81.6
C-T	81.4	87.3	63.1	35.0	36.4	29.0	32.7	36.8	28.6	37.5	32.6	28.8	36.4
F	253.4	346.8	162.3	138.2	166.4	165.0	140.4	172.0	148.5	126.7	143.8	124.5	136.8
D-T	25.1	20.8	21.3	13.7	22.6	18.6	22.7	16.9	19.5	23.4	17.2	22.6	18.8
F	35.6	90.8	105.9	56.4	113.4	78.2	96.4	68.5.	86.8	125.2	72.9	94.4	81.3
E-T	51.7	44.2	37.6	50.0	37.1	36.3	32.8	37.6	42.4	48.7	36.9	43.2	36.5
F	137.9	180.8	169.6	191.9	169.3	193.3	182.0	171.9	164.1	160.3	174.5	187.6	168.4
F-T	12.7	112.2	10.7	3.6	12.5	13.8	9.8	7.6	8.7	10.4	12.5	11.8	10.6
F	−15.0	442.3	457.0	17.5	74.6	79.8	57.2	43.3	51.6	59.9	73.4	65.8	61.2
G-T	232.0	9.6	12.8	15.0	16.5	15.8	16.8	14.6	16.2	18.7	20.5	22.5	24.8
F	324.6	12.8	16.4	18.7	21.2	20.8	21.6	18.8	20.5	22.4	23.8	25.2	27.4
H-T	216	8.8	10.7	13.6	15.6	16.7	17.8	16.4	14.6	17.3	19.3	21.8	23.5
F	228.0	10.6	14.4	17.5	20.8	22.4	22.8	20.2	19.6	21.8	22.9	24.6	26.5

Table A4. Distribution of banks by their traditional and equity—sset ratios for 2008–2020 (%)

Bank	Ratio	08	09	10	11	12	13	14	15	16	17	18	19	20
A-T	($\frac{TE}{TA}$)%	9.0	11.6	5.3	5.3	11.2	15.5	16.7	20.7	23.4	23.6	25.0	28.9	31.7
F	($\frac{FE}{TA}$)%	6.8	4.8	0.5	3.7	7.2	6.8	6.4	7.8	7.5	6.6	7.6	8.5	7.8
B -T	($\frac{TE}{TA}$)%	10.0	9.7	11.8	14.8	12.5	11.9	11.4	12.7	12.8	15.1	15.4	17.3	17.3
F	($\frac{FE}{TA}$)%	4.2	5.2	4.3	5.4	5.0	7.3	4.6	5.4	4.9	6.7	5.8	6.4	6.3
C-T	($\frac{TE}{TA}$)%	2.3	5.4	0.9	9.1	12.8	7.7	9.2	10.3	7.8	9.8	7.1	9.5	10.3
F	($\frac{FE}{TA}$)%	0.3	0.1	0.9	1.6	1.5	2.0	2.8	2.2	3.5	4.6	3.2	4.3	4.8
D-T	($\frac{TE}{TA}$)%	9.5	10.1	10.9	17.2	10.6	13.3	12.4	9.6	9.5	10.2	10.1	11.4	12.3
F	($\frac{FE}{TA}$)%	8.6	5.0	4.7	11.2	3.9	4.2	4.8	7.6	6.4	4.1	4.3	7.2	5.7
E-T	($\frac{TE}{TA}$)%	3.8	4.8	6.5	5.0	7.4	8.5	11.8	7.4	11.0	9.9	9.4	10.8	10.3
F	($\frac{FE}{TA}$)%	1.6	0.9	1.7	1.3	1.2	1.8	2.5	1.4	2.3	1.9	2.4	3.2	2.8
F-T	($\frac{TE}{TA}$)%	12.8	2.5	14.4	7.1	23.6	11.3	12.5	10.6	13.4	13.9	11.6	12.2	15.3
F	($\frac{FE}{TA}$)%	8.0	0.10	0.10	8.8	26.9	9.3	8.6	12.4	0.80	0.72	8.8	7.5	10.7
G-T	($\frac{TE}{TA}$)%	0.38	0.64	0.74	0.63	0.40	0.66	0.61	0.65	0.63	0.61	0.61	0.62	0.58
F	($\frac{FE}{TA}$)%	0.32	0.60	0.68	0.58	0.36	0.62	0.55	0.58	0.58	0.55	0.55	0.56	0.52
H-T	($\frac{TE}{TA}$)%	0.46	0.60	0.37	0.23	0.21	0.20	0.17	0.17	0.17	0.15	0.15	0.13	0.12
F	($\frac{FE}{TA}$)%	0.42	0.56	0.33	0.18	0.17	0.16	0.16	0.16	0.16	0.14	0.14	0.12	0.11