## Monday, June 24, 2019

### Statistical Tests for Debt Equity Versus Debt Assets

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 non-parametric 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 mid-cap growth fund) and for 20 firms in VOE (Vanguard mid-cap 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 non-parametric 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

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 t-test D/E VOT-VOE equal variance 0.133 t-test D/A VOT-VOE equal variance 0.004 t-test D/A not equal variance 0.004

The two-tailed t-test 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:

Non-parametric 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)

Two-sample Wilcoxon rank-sum (Mann-Whitney) test

Type |      obs    rank sum    expected
-------------+---------------------------------
1 |       20         309         410
2 |       20         511         410
-------------+---------------------------------
combined |       40         820         820

----------

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)

Two-sample Wilcoxon rank-sum (Mann-Whitney) test

Type |      obs    rank sum    expected
-------------+---------------------------------
1 |       20         309         410
2 |       20         511         410
-------------+---------------------------------
combined |       40         820         820

----------

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:

Mid-cap value firms have more debt than mid-cap growth firms in these samples.

Statistical tests and models based on debt equity ratios can provide misleading results.