This post compares two leverage statistics  the debt equity ratio and the debt asset ratio  for value and growth stocks. The debt equity ratio is more highly skewed than the debt asset ratio. The results presented here demonstrate the value of using a nonparametric
test when some data is highly skewed.
Question: Below are the debt equity D/E and debt asset
D/A ratios for 20 firms in VOT (Vanguard midcap growth fund) and for 20 firms
in VOE (Vanguard midcap value fund).
Analysis based on
classical statistics:
What are the median, mean
standard deviation range, interquartile range, and standard deviation skew and
kurtosis for the growth stocks and for the value stocks?
Does a classical test on
difference in D/E ratio between growth and value lead to the same result as a classical
test in D/A ratio between growth and value stocks?
Analysis based on
nonparametric statistics:
Why will the rank of D/E and
rank of D/A be identical for the combined sample?
Conduct the Wilcoxon Rank sum
tests for difference in DE between growth and value stocks and difference in
rank sum tests for DA.
Show that the Wilcoxon Rank
sum test results are identical regardless of whether the test was done on D/E
or D/A?
The Data:
Debt Equity Vs Debt Assets for Midcap Growth and Value Firms


Type

Obs

DE

DA

Growth Firms


1

1

4.76

0.83

1

2

2.77

0.73

1

3

2.72

0.73

1

4

0.93

0.48

1

5

1.08

0.52

1

6

1.51

0.60

1

7

0.55

0.35

1

8

15.36

0.94

1

9

1.52

0.60

1

10

1.27

0.56

1

11

1.07

0.52

1

12

0.47

0.32

1

13

0.54

0.35

1

14

0.92

0.48

1

15

2.19

0.69

1

16

0.82

0.45

1

17

0.35

0.26

1

18

5.00

0.83

1

19

2.13

0.68

1

20

0.18

0.15

Value Firms


2

21

7.05

0.88

2

22

3.64

0.78

2

23

6.46

0.87

2

24

2.05

0.67

2

25

1.53

0.61

2

26

6.28

0.86

2

27

3.66

0.79

2

28

2.28

0.70

2

29

6.67

0.87

2

30

0.94

0.49

2

31

2.28

0.69

2

32

2.21

0.69

2

33

2.33

0.70

2

34

6.51

0.87

2

35

15.69

0.94

2

36

1.43

0.59

2

37

1.08

0.52

2

38

3.71

0.79

2

39

2.56

0.72

2

40

0.88

0.47

Classical Statistics
Growth Stock Statistics


DE

DA


Mean

2.31

0.55

Median

1.18

0.54

STD

3.35

0.21

Min

0.18

0.15

Max

15.36

0.94

25th

0.75

0.43

75th

2.32

0.70

skew

3.43

0.05

kurt

13.18

0.51

Value Stock Statistics


DE

DQ


Mean

3.96

0.72

Median

2.44

0.71

STD

3.47

0.14

Min

0.88

0.47

Max

15.69

0.94

25th

1.92

0.66

75th

6.32

0.86

skew

2.19

0.38

kurt

6.18

0.76

Comments on Classical Stock
Statistics:
value stocks appear more
highly leveraged than growth stocks.
The skew and standard deviation
of debt equity statistics is huge. This
will likely affect hypothesis tests based on the debt equity ratio.
Classical Hypothesis Tests:
Hypothesis Tests for Difference in Mean Ratios


P Value


ttest D/E VOTVOE equal variance

0.133

ttest D/A VOTVOE equal variance

0.004

ttest D/A not equal variance

0.004

Comments:
The twotailed ttest for no
difference between leverage of growth and value stocks is rejected only for the
debt asset ratio.
The skew of the Debt equity
ratio makes classical tests invalid.
Observing the ranks of the
combined sample:
Nonparametric Tests:
DE and DA will have the same
rank order. This follows because Assets
is equal to Equity + Debt. If D over E
is relatively high then D over E+D must be relatively high.
The Wilcoxon rank sum test is
a comparison of the sum of ranks from the two groups in a combined sample.
Here are the results for the
debt equity ratio.
ranksum de, by(Type)
Twosample Wilcoxon ranksum
(MannWhitney) test
Type  obs
rank sum expected
+
1  20 309 410
2  20 511 410
+
combined 
40 820 820
unadjusted variance 1366.67
adjustment for ties 0.00

adjusted variance 1366.67
Ho: de(Type==1) = de(Type==2)
z = 2.732
Prob > z = 0.0063
Here are the results for Debt
Assets
. ranksum da, by(Type)
Twosample Wilcoxon ranksum
(MannWhitney) test
Type  obs
rank sum expected
+
1  20 309 410
2  20 511
410
+
combined  40 820 820
unadjusted variance 1366.67
adjustment for ties 0.00

adjusted variance 1366.67
Ho: da(Type==1) = da(Type==2)
z = 2.732
Prob > z = 0.0063
Comment:
The Wilcoxon results are
identical regardless of whether one uses Debt Equity or Debt Assets.
Conclusions:
Midcap value firms have more
debt than midcap growth firms in these samples.
Statistical tests and models
based on debt equity ratios can provide misleading results.
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