Coefficient of Variation with negative values

[sleepybird]

Hello,
I want compare earning volatility of company x with other companies. I intend to measure earning volatility by calculating coefficient of variation of net earnings. As you know, net earnings can be both positive and negative, so my question is should I take the absolute value of the net earnings for the calculation or CV simply doesn't work in this case. Thanks.

 

[David Harper CFA FRM]

Hi @sleepybird good to see you back I've never in practice used the coefficient of variation. Of course, mathematically (aside from zero mean), there is nothing that seems require an adjustment to (σ/µ) of a series, if the series happens to contain negatives. As i'm sure you know the numerator's volatility already squares the deviations, so the numerator doesn't present any obvious issue. But you raise an interesting point. If i just try on four series (n = 5, to keep it super simple):

• scenario #1: {0, 1, 2, 3, 4} such that (I am using pop sd to keep it simple, I'd imagine sample sd is better but it doesn't matter for this): µ = 2.0, σ = 1.4 and σ/µ = 0.7
• scenario #2: {6, 7, 8, 9, 10} such that µ = 8.0, σ = 1.4 and σ/µ = 0.2; I chose this to illustrate COV "does its job:" sigma is identical but COV in #2 is appropriately scaled to the higher mean
  
• scenario #3: {-1, 0, 1, 2, 3} such that µ = 1.0, σ = 1.4 and σ/µ = 1.4; shifting all items by only -1 doubles the COV feels wrong
  
• scenario #4: {-4, 1, 1, 2, 3} such that µ = 0.2, σ = 2.5 and σ/µ = 2.5; really jumps, also feels wrong

If we take absolute values of #3 and #4 per your idea, I get COV(#3) = 0.7 which feels better, and COV(#4) = 0.5 = 1.2/2.2 which sort of does not. All I can say is that, personally, neither applications to #3 or #4 resonate for me. I just gave this a few minutes thought, but I can't see myself see any fix to the "near axis" (i.e., where denominator is influenced toward zero by negative values) implementation of COV ... This would not be the first time a metric doesn't work when the series "crosses the axis." Thanks,

 

[sleepybird]

@David Harper CFA FRM, appreciate your response. First let me just say that BT is such a great website for resource and thanks for helping me passing the exams in 2013!

I was reading this: http://faculty.tuck.dartmouth.edu/images/uploads/faculty/robert-resutek/DR3_0913.pdf. On page 23: “The coefficient of variation scales earnings uncertainty by the absolute value of expected earnings, thereby preventing firms with extreme earnings (and generally higher earnings volatility) from disproportionately populating the extremes of the earnings uncertainty distribution.” This gives me the impression we should take the absolute values of the earnings.

If you think CV is not a good measure due to negative values, what other measures would you recommend?

If the mean is causing the issue for the CV calculation, then should we simply use standard deviation? I had thought about this before, but the issue is that if I compare a small company with a large company, the standard deviation is not really comparable. For example, for company X (bigger), I get σ = 388, whereas for company Y (smaller), σ = 96. I don’t think I can say company X is more volatile, correct?

What should I do to solve this issue? I thought about standardizing the standard deviation. I used the z-score, but this makes σ = 1 for all companies. What am I missing here?

 

[David Harper CFA FRM]

Hi @sleepybird Thank you for the kind words The paper is interesting thanks. I did not mean to say CV is wholesale a bad measure. I've been practicing data science for years now, and I don't think i've seen it used but a few times. Personally, I suspect the reason is simply that we rarely depend on a single measure to bear too much load. The standard deviation is illustrated along with several other summary measures; standard deviation itself being but one measure of variability. If you think about a distribution, we know from the FRM that the standard deviation is just (a function of) one moment: a distribution needs more descriptors. Similarly, with company earnings analysis, it would be complemented by several metrics (this reminds me of EVA which i worked with extensively years ago. The folks behind EVA were always trying to position it as the single metric that subsumed all important information, but practitioners just never quite bought it, in my experience high level practitioners do not put too much weight on any single metric could characterize company performance).

So I guess my thoughts are:

• The CV remains useful in general. If I were using it and reporting it, I might caveat (asterisk) its sensitivity near the axis (near the zero). Based on my quick look above, I actually perceive the issue more around zero axis (which is not uncommon, it's an issue with mere returns). Not so much negatives per se, but it looks too sensitive as the average approaches zero. No biggie, data is rarely perfect, I'd handle those "near axis" cases with an asterisk.
  
• In my naive impression, the absolute value "fixes" a certain weakness but might create other issues (e.g, I just tried series -10,-5,1,7, 12, such that absolute values chop the σ and COV in half and I'm not sure I like that outcome?).
• If I really wanted to scale by company size, I might try σ(earnings)/(average company sales), or for that matter σ(earnings)/(some other proxy for size); I've never seen this, so please don't think i've considered it carefully, but off the cuff, this appeals to be if the goal really is "scaling earning volatility according to size differences". I hope my thoughts are helpful!

 

[sleepybird]

@David Harper CFA FRM,, thanks for your quick response. Very helpful.
I guess I can use CV in my case as the average earnings in my data set are not close to zero (no zero axis issue).
Also thanks for suggestions on scale by company size. Maybe I can complement CV with that measure as well.
Great point on that no single metric is perfect.