CHAPTER VI - STATISTICAL MODELS AND TESTS
6.5 Wilcoxon Signed Rank test – practical example
6.5.1 Company specific information
The Wilcoxon Signed Rank test is not a commonly known statistical model as e.g. a regression analysis. To provide the reader with a basic understanding of how it works in practice and not just the theory behind, we have included a practical example with illustrations and comments on the modelling and calculations. The example will be based upon the portfolio company
‘Color Print A/S’ and the test of EBITDA-margin development for the entire sample.
Our data is manually entered into our Excel model based on annual reports from the portfolio companies. The first part of the company information is general information such as name
5 If the EBITDA-margin increases from 1% in 𝐸𝑛𝑡𝑟𝑦−1 to 2% in 𝐸𝑥𝑖𝑡+1 it is an increase on only 1%point but on 100%. Contradictory, a change in EBITDA-margin from 10% in 𝐸𝑛𝑡𝑟𝑦−1 to 15% in 𝐸𝑥𝑖𝑡+1is an increase on 5%
points but only on 50%. Even though the companies performance is significantly better on the second case, the percentage change is not accordingly demonstrating it, which is why the percentage point measure is used on ratios.
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and CVR (the Danish central business register), which is used for our automatized modelling formulas to reduce manual test procedures. The industry code is how we technically match the companies’ development to its peers, to isolate the company specific development and remove the industry & market movements. In this case, Color Print A/S is matched with industry code 18000 “Printing” and will be adjusted by the the printing industries average.
Income statement information is entered from the year prior to entry and the year after exit.
Balance sheet information is entered similarly to the income statement information, and in addition two balance sheet items used to calculate ratios are also entered from two year prior to entry and on the year for exit to obtain yearly averages.
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Table 9: Step 1 - Accounting information (Color Print A/S example)
Source: Own contribution
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6.5.2 Company sampling, industry figures and statistical setup
All the portfolio companies are listed on a new Excel-sheet where the statistical test is performed. The CVR code is used as base for Excel to recognize which Excel sheet to draw information from, to reduce manual entering. The EBITDA-margin growth is -6.9% for Color Print A/S because of a decrease from 12,9% in 2005 to 6,0% in 2013. The -6,9% will be industry adjusted to determine Color Print’s growth adjusted for market and industry movements.
Table 10: Step 2 - Initial calculations for the statistical test
Source: Own contribution
CVR Navn Industri-nr. e-1 e+1 EBITDA e-1 EBITDA e+1 EBITDA growth Sector
17523716 AS Cimbria 28002 2006 2013 6,2% 9,5% 3,2% Industrials
17260839 Biblioteksmedier 58001 2007 2014 2,6% 11,3% 8,7% Service
43059211 Bodilsen 31000 2005 2008 -4,2% -6,4% -2,1% Industrials
34281718 BTX Group A/S (Brandtex) 46004 2004 2014 3,0% 10,8% 7,8% Consumer goods 78804912 C&D Foods A/S (Arovit Petfood) 10005 2005 2009 9,1% 2,1% -7,0% Other
12516479 Chr. Hansen A/S 20001 2004 2011 13,0% 31,7% 18,7% Healthcare
10638178 Color Print A/S 18000 2005 2013 12,9% 6,0% -6,9% Service
21474878 CRH Concrete A/S (H+H Fiboment)23002 2003 2008 4,2% 0,5% -3,7% Industrials
33211317 DAKO Danmark A/S 32001 2006 2014 -20,9% 7,6% 28,5% Healthcare
56416218 Damixa ApS 28002 2008 2014 -14,4% 4,5% 18,9% Industrials
19778649 DK Company Online A/S (Bon' A Parte)47008 2009 2014 -5,4% -14,9% -9,5% Consumer goods
84667811 EG A/S 62000 2007 2014 6,9% 10,9% 4,0% TMT
66920216 F. Junckers Industrier A/S 16000 2003 2014 20,8% 3,0% -17,9% Industrials
13246092 Ferrosan A/S 21000 2004 2012 15,9% -46,2% -62,2% Healthcare
33487819 Fibervisions A/S 13000 2005 2013 0,4% 1,6% 1,1% Industrials
67287118 Glud & Marstrand A/S 24000 2000 2006 7,1% 7,0% -0,1% Industrials
12006632 Gram Commercial A/S 28002 2004 2009 3,0% 7,8% 4,8% Industrials
89523818 Haarslev Industries A/S 28002 2005 2013 13,5% 12,0% -1,6% Industrials
10161614 ISS 81000 2004 2014 7,2% 6,6% -0,6% Service
10276802 Jeld-Wen A/S (Vest-Wood) 16000 2001 2006 11,7% 4,6% -7,1% Industrials
26911745 KMD A/S 62000 2008 2013 8,9% 9,4% 0,5% TMT
40559817 Kompan A/S 32002 2004 2012 3,8% 8,9% 5,1% Industrials
25690869 Kosan Crisplant A/S 28002 2003 2009 13,6% 0,7% -12,8% Industrials
13233934 Kuben Ejendomsadministration A/S81000 2007 2010 -8,8% 4,3% 13,1% Service
14707204 Københavns Lufthavne A/S 52000 2004 2012 58,3% 63,9% 5,5% Other
36887613 Legoland ApS 55000 2004 2014 19,6% 20,2% 0,7% Other
21330248 Logstor A/S 24000 2005 2014 10,8% 5,1% -5,7% Industrials
86479915 Mediq Danmark A/S (Kirudan) 46007 2005 2008 4,9% 7,3% 2,4% Healthcare
21783684 Netcompany A/S 62000 2005 2012 20,3% 25,8% 5,5% TMT
17484575 Novasol A/S (Dansommer) 68002 2000 2003 5,4% 5,2% -0,1% Service
16926647 Novenco Building & Industry A/S (Novenco Group)28002 2005 2014 -6,1% -21,0% -14,9% Industrials
25786823 OW Bunker 46007 2006 2012 1,3% 0,8% -0,5% Other
17557491 Rahbekfisk A/S 10002 2004 2009 5,1% 0,5% -4,6% Industrials
14820701 Scandinavian Business Seating A/S31000 2006 2014 9,8% 12,9% 3,2% Industrials
38978217 Schades A/S 17000 2008 2014 2,2% 0,4% -1,8% Industrials
41333715 Skamol A/S 23001 2006 2013 4,6% 23,7% 19,1% Industrials
25141350 Sonion A/S 32001 2008 2014 -5,9% 6,1% 11,9% Healthcare
42405310 Stofa A/S 61000 2008 2013 17,5% 20,4% 2,9% TMT
14773908 TDC 61000 2005 2014 27,8% 41,9% 14,1% TMT
19989348 TIA Technology A/S 62000 2005 2014 34,0% -5,8% -39,7% TMT
11810543 Unifeeder A/S 50000 2006 2014 5,9% 7,0% 1,1% Other
64153315 Unomedical A/S 32001 2002 2009 21,0% 8,4% -12,6% Healthcare
61094628 Xellia Pharmaceuticals 21000 2007 2014 15,5% -12,3% -27,9% Healthcare
EBITDA-Margin Company figures
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The portfolio companies are divided into 6 major industries to test for industry differences in the extended analysis by the Kruskal-Wallis H-test.
The industry figures are paired with the company’s CVR code and the specific industry no., assigned to all portfolio companies to compare each portfolio company with the correct industry average. The industry numbers are accordingly extracted from the REGN1 database, in this example the EBITDA. In a separate sheet we have calculated the EBITDA-margin by dividing the EBITDA with its corresponding revenue. Columns 2, 3, 5 & 6 in table 11 are formulas used to extract the correct EBITDA-margin value from table 12. In Color Prints’
(CVR: 10638178) case the EBITDA-margin for the Printing Industry in 2005 (entry year) is in column 6 – row 6 (column begins in the 2000 column), in table 12, which is 12,6%.
Table 11: Step 3 – Adjusted growth calculation and Wilcoxon Test
Source: Own contribution
CVR EBITDA growth Diff. Sign |Diff.| Rank
17523716 12 7 8,9% 12 14 7,8% -1,1% 4,4% 4,4% + 4% 16
17260839 23 8 5,0% 23 15 8,4% 3,4% 5,3% 5,3% + 5% 21
43059211 13 6 11,7% 13 9 7,5% -4,2% 2,1% 2,1% + 2% 8
34281718 16 5 5,7% 16 15 6,9% 1,2% 6,6% 6,6% + 7% 23
78804912 2 6 7,4% 2 10 5,4% -2,0% -5,0% -5,0% - 5% 19
12516479 7 5 23,4% 7 12 26,4% 3,0% 15,7% 16% + 16% 35
10638178 6 6 12,6% 6 14 9,0% -3,6% -3,3% -3,3% - 3% 11
21474878 10 4 13,0% 10 9 12,8% -0,2% -3,4% -3,4% - 3% 12
33211317 14 7 12,8% 14 15 17,4% 4,6% 23,9% 23,9% + 24% 40
56416218 12 9 7,1% 12 15 8,0% 0,9% 18,0% 18,0% + 18% 37
19778649 19 10 4,8% 19 15 4,4% -0,4% -9,0% -9,0% - 9% 30
84667811 25 8 10,3% 25 15 11,2% 0,9% 3,1% 3,1% + 3% 10
66920216 4 4 8,4% 4 15 8,1% -0,2% -17,6% -18% - 18% 36
13246092 8 5 23,3% 8 13 28,0% 4,7% -66,8% -66,8% - 67% 43
33487819 3 6 10,0% 3 14 8,2% -1,8% 2,9% 2,9% + 3% 9
67287118 11 1 7,4% 11 7 11,1% 3,6% -3,8% -3,8% - 4% 13
12006632 12 5 6,9% 12 10 2,9% -3,9% 8,8% 8,8% + 9% 29
89523818 12 6 7,3% 12 14 7,8% 0,4% -2,0% -2,0% - 2% 7
10161614 27 5 14,6% 27 15 12,4% -2,2% 1,6% 1,6% + 2% 5
10276802 4 2 9,1% 4 7 9,6% 0,4% -7,5% -7,5% - 8% 26
26911745 25 9 10,1% 25 14 10,8% 0,7% -0,2% 0% - 0% 1
40559817 15 5 7,9% 15 13 33,1% 25,2% -20,1% -20,1% - 20% 38
25690869 12 4 6,1% 12 10 2,9% -3,1% -9,7% -9,7% - 10% 31
13233934 27 8 13,4% 27 11 13,9% 0,6% 12,5% 12,5% + 13% 33
14707204 21 5 8,1% 21 13 5,7% -2,3% 7,9% 7,9% + 8% 28
36887613 22 5 13,0% 22 15 11,9% -1,1% 1,7% 1,7% + 2% 6
21330248 11 6 6,0% 11 15 6,2% 0,2% -5,9% -5,9% - 6% 22
86479915 18 6 5,3% 18 9 3,0% -2,3% 4,7% 4,7% + 5% 18
21783684 25 6 11,0% 25 13 9,8% -1,2% 6,7% 7% + 7% 25
17484575 26 1 54,7% 26 4 55,2% 0,5% -0,7% -0,7% - 1% 4
16926647 12 6 7,3% 12 15 8,0% 0,7% -15,6% -15,6% - 16% 34
25786823 18 7 3,8% 18 13 3,1% -0,7% 0,2% 0,2% + 0% 3
17557491 1 5 5,0% 1 10 5,1% 0,0% -4,6% -4,6% - 5% 17
14820701 13 7 10,5% 13 15 9,6% -1,0% 4,1% 4,1% + 4% 15
38978217 5 9 4,6% 5 15 8,0% 3,4% -5,2% -5,2% - 5% 20
41333715 9 7 10,9% 9 14 9,2% -1,7% 20,8% 20,8% + 21% 39
25141350 14 9 12,1% 14 15 17,4% 5,3% 6,6% 7% + 7% 24
42405310 24 9 27,5% 24 14 30,2% 2,7% 0,2% 0,2% + 0% 2
14773908 24 6 22,3% 24 15 28,5% 6,2% 7,9% 7,9% + 8% 27
19989348 25 6 11,0% 25 15 11,2% 0,2% -39,9% -39,9% - 40% 42
11810543 20 7 4,6% 20 15 9,7% 5,0% -3,9% -3,9% - 4% 14
64153315 14 3 10,7% 14 10 10,2% -0,5% -12,1% -12,1% - 12% 32
61094628 8 8 19,5% 8 15 29,6% 10,1% -37,9% -37,9% - 38% 41
Industry figures Adjusted
growth
Wilcoxon test EBITDA-margin e-1 EBITDA-margin e+1
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Column 2, 3, 5 & 6 in table 11 are thus just formulas made to reduce human errors by manually finding and typing the numbers into the test. This is done for both the year prior to entry and the year after exit, to find the industry’s development within the same time frame as the portfolio companies holding period. The printing industry average EBITDA-margin development is a 3.6% decrease and with Color Print’s 6,9% decrease its industry-adjusted development is a 3,3% decrease.
Table 12: Step 4 - Industry averages EBITDA-margin
Source: Own contribution
The industry-adjusted growth for each portfolio is hereafter divided into either an increase or a decrease, and assigned either + or -. Each industry-adjusted EBITDA-margin development is hereafter ranked by their absolute growth, without taking into account whether it is a decrease or an increase. The smallest absolute growth (often around 0), will receive the rank 1 and so forth until the largest absolute growth (CVR:13246092) with an industry-adjusted EBITDA-margin decrease on -66,8% receives the rank 43. Color Print A/S is ranked 11, with an absolute development on 3,3%.
6.5.3 Statistical test conclusions
When performing the test, the sign of the development is important again, because all observations with a negative sign is summed and similarly for the positive signs. In this case, the sum of ranks for positive observations is 453 and the sum of ranks for negative observations is 493. The smallest of these observations, in this case the sum of positive ranks, is equal to the test size. The test size is compared to the critical value to determine whether or not the test has any significance. If the test size is larger than the critical value, the test shows no sign of significance, and we cannot conclude that the PE-owned portfolio companies
Industry average: EBITDA-margin Indust.code 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
Fiskeindustri 10002 4,9% 6,1% 6,5% 5,4% 5,0% 4,0% 3,4% 2,2% 3,4% 5,1% 4,6% 3,7% 3,0% 2,0% 3,6%
Anden fødevareindustri 10005 7,4% 8,4% 8,4% 6,9% 7,1% 7,4% 7,1% 6,3% 4,5% 5,4% 7,0% 6,2% 6,2% 6,8% 6,3%
Tekstilindustri 13000 9,9% 10,4% 11,1% 10,5% 10,4% 10,0% 9,0% 7,9% 8,1% 5,9% 8,3% 7,8% 8,0% 8,2% 9,4%
Træindustri 16000 9,7% 9,1% 8,8% 8,4% 9,2% 10,7% 9,6% 9,0% 5,1% 3,9% 6,0% 5,9% 6,7% 7,6% 8,1%
Papirindustri 17000 12,0% 11,3% 11,1% 11,6% 8,9% 7,7% 8,0% 6,1% 4,6% 5,8% 5,8% 8,3% 7,0% 8,6% 8,0%
Trykkerier mv. 18000 12,0% 11,9% 12,3% 12,1% 13,8% 12,6% 12,4% 11,5% 8,4% 8,1% 8,6% 9,9% 9,1% 9,0% 8,8%
Fremst. af basiskemikalier 20001 18,2% 17,3% 16,6% 16,7% 23,4% 23,6% 20,5% 13,8% 21,7% 19,2% 28,1% 26,4% 25,2% 23,9% 25,9%
Medicinalindustri 21000 27,3% 26,6% 23,1% 25,3% 23,3% 22,5% 19,4% 19,5% 24,4% 20,9% 22,8% 24,7% 28,0% 28,6% 29,6%
Glasindustri og keramisk industri 23001 10,2% 5,2% 9,6% 7,5% 7,4% 11,1% 10,9% 13,7% 9,4% 1,8% 6,7% 8,6% 7,6% 9,2% 10,8%
Betonindustri og teglværker 23002 16,5% 13,7% 13,6% 13,0% 11,8% 12,9% 13,6% 14,7% 12,8% 10,1% 7,0% 10,8% 10,3% 10,4% 11,1%
Fremst. af metal 24000 7,4% 6,9% 8,4% 6,8% 9,9% 6,0% 11,1% 10,8% 7,8% -1,7% 6,6% 5,2% 3,9% 4,1% 6,2%
Fremst. af andre maskiner 28002 6,4% 7,0% 6,9% 6,1% 6,9% 7,3% 8,9% 9,0% 7,1% 2,9% 5,2% 7,7% 8,3% 7,8% 8,0%
Møbelindustri 31000 10,8% 11,4% 11,6% 11,7% 12,3% 11,7% 10,5% 9,9% 7,5% 6,8% 8,9% 8,6% 9,5% 9,5% 9,6%
Fremstilling af medicinske instrumenter mv. 32001 13,3% 12,0% 10,7% 11,4% 11,3% 15,1% 12,8% 13,8% 12,1% 10,2% 19,9% 13,7% 15,6%#VALUE! 17,4%
Legetøj og anden fremstillingsvirksomhed 32002 5,1% 10,2% 13,2% 4,3% 7,9% 4,8% 10,1% 15,4% 28,6% 30,3% 32,0% 31,3% 33,1%#VALUE! #VALUE!
Engrosh. med tekstiler og husholdningsudstyr 46004 5,1% 5,0% 5,2% 5,5% 5,7% 7,1% 6,3% 7,0% 5,9% 4,7% 6,8% 5,8% 5,5% 6,2% 6,9%
Engrosh. med it-udstyr 46005 4,4% 2,6% 2,9% 2,9% 3,3% 2,7% 2,4% 3,2% 3,5% 3,5% 4,6% 4,1% 4,0% 3,9% 5,1%
Anden engroshandel 46007 4,8% 4,4% 4,9% 4,4% 5,5% 5,3% 3,8% 4,4% 3,0% 3,2% 3,6% 3,1% 3,1% 3,3% 3,8%
Internethandel, postordre mv. 47008 5,3% 4,5% 7,6% 7,0% 8,6% 6,1% 8,3% 8,1% 6,9% 4,8% 7,5% 4,5% 5,3% 5,0% 4,4%
Skibsfart 50000#VALUE! 10,0% 11,1% 14,8% 23,0% 17,8% 4,6% 9,4% 8,1% -6,5% 12,1% 3,7% 2,9% 6,4% 9,7%
Hjælpevirksomhed til transport 52000 7,4% 6,5% 6,4% 7,8% 8,1% 7,1% 6,5% 8,4% 7,2% 5,9% 5,9% 5,3% 5,7% 6,2% 6,4%
Hoteller mv. 55000 17,1% 14,5% 14,8% 13,0% 13,0% 13,5% 14,2% 16,3% 13,9% 9,7% 8,7% 9,5% 11,1% 11,8% 11,9%
Forlag 58001 9,8% 8,9% 8,0% 8,2% 10,3% 13,3% 8,4% 5,0% 13,4% 3,4% 7,0% 6,7% 5,9% 8,5% 8,4%
Telekommunikation 61000 -4,7% 22,0% 20,3% 22,3% 20,7% 22,3% 24,7% 24,7% 27,5% 30,7% 28,8% 29,5% 28,4% 30,2% 28,5%
It-konsulenter mv. 62000 4,8% 5,9% 7,2% 8,2% 11,1% 11,0% 11,2% 10,3% 10,1% 10,5% 9,7% 9,7% 9,8% 10,8% 11,2%
Boligudlejning 68002 54,7% 60,3% 56,1% 55,2% 48,2% 50,2% 86,5% 28,4% 19,1% 43,2% 39,1% 47,1% 44,1% 39,4% 41,6%
Ejendomsservice, rengøring og anlægsgartnere 81000 14,8% 16,4% 15,8% 14,6% 14,6% 13,5% 14,1% 13,4% 12,7% 13,2% 13,9% 13,3% 12,8% 11,9% 12,4%
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outperform or underperform relative to their peers. If the direction on test level shows a decrease, the sum of ranks for negative observations is higher than the sum of ranks for positive observations, which indicates that the portfolio companies underperform relative to their industry peers. However, to conclude anything the test output must be significant, which it in this case, is far from.
Table 13: Step 5 – Final test statistics on EBITDA margin cf. Wilcoxon
Source: Own contribution
For a total output of the statistical tests for all portfolio companies included in our sample, we refer to appendix 5 and 6.
With chapter 1-6 in mind, we will in the following chapter present, analyse and discuss all the results from running the statistical tests cf. Wilcoxon Signed Rank test and Kruskal-Wallis H-test.
Test of EBITDA Margin
Observations 43
Alpha (significance) 0,05
Mean of ranks 473
Std.dev of ranks 82,82
Sum of ranks, positive (+) 453 Sum of ranks, negative (-) 493 Mean of ranks, positive (+) 20,59 Mean of ranks, negative (-) 21,57
Critical value 310,63
Test size 453
Direction Decrease
Significance No
Z-size -0,2415
P-value 0,8092
Middle -2,53%
Median 0,21%
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