Percent Changes?

Benjamin Kay writes about a problem that seems simple but actually is not:

I’ve come across a pair of problems in my work into which you may have some insight. I am looking at the percentage change in earnings per share (EPS) of various large American companies over a 3 year period. I am interested in doing comparisons of how other attributes influence the median value of earnings per share. For example, it might be that high paying companies have higher EPS growth than low paying ones. I am aware that this model might not fully take advantage of the data but I’m preparing it for an audience with limited statistical education.

The problems occur in ranking percentages. If you calculate percentages as (New – Old)/Old then there are two major problems:
1) Anything near zero explodes
2) Companies which go from negative to positive EPS appear to have negative growth rates. (1$) -(- $1) / -$1 = -200%

The first problem is seemingly intractable as long as I am using percent changes, but I cannot use dollar changes because it ignores the issue of scale. A company with 100 shares and $100 in earnings has $1 EPS, and one with 20 shares (and the same earnings) has $5 EPS. If both companies double their earnings to $200 dollars, they’ve performed identically. However, in absolute changes the former shows $1 change and the latter $5. I’m stuck with what to do here, maybe there is another measure of change that I haven’t considered or another way of doing this entirely.

One thing I’ve considered for the second problem is taking the absolute value of companies whose EPS changes sign. That seems equivalent to claiming that a change from $1 to $3 EPS is equivalent to a -$1 to $1 change in EPS. Is that a standard approach to treating percent changes? Are there any other assumptions lurking underneath when doing this?

Is there a classic reference to doing order statistic work like this on percentile data?

My reply: this is an important problem that comes up all the time. The percent-change approach is even worse than you suggest, because it will blow up if the denominator approaches zero. Similar problems arise with marginal cost-benefit ratios, LD50 in logistic regression (see chapter 3 of Bayesian Data Analysis for an example), instrumental variables, and the Fieller-Creasy problem in theoretical statistics. I’ve actually been planning for awhile to write a paper on estimation of ratios where the denominator can be positive or negative.

In general, the story is that the ratio completely changes in interpretation when the denominator changes sign (as you illustrated in your example). But yeah, dollar values can’t be right either. I have a couple questions for you:

a. How important are the signs to you? For example, if a given company changes from -$1 to $1, is that more important to you than a change from $1 to $3, or from $3 to $5?

b. For any given company, do you want to use the same scaling for all three years? I imagine the answer is Yes (so you don’t have to worry about funny things happening such as an increase of 25%, followed by a decrease of 25%, does not bring things to the initial value).

One approach might be to rescale based on some relevant all-positive variable such as total revenue. I’m sure many other good options are available, once you get away from trying to rescale based on a variable that can be positive or negative.

5 thoughts on “Percent Changes?

  1. The same problem comes up with interest rates. In many cases using 1+r rather than r (where r is the interest rate) is both theoretically more justifiable and computationally better behaved. Earnings per share price (rather than earnings per share) is somewhat analogous to an interest rate, so it might be possible to use 1+earnings/price. If the variability in price is a problem, perhaps use a smoothed price.

  2. Would it make sense to include the first difference of the numerator on the LHS and the denominator on the RHS? The denominator could also be interacted with other covariates on LHS as theory/exploratory analysis suggests. You would be asking a slightly different question, but it looks like you have two varying components in a ratio, the numerator and the denominator, and by collapsing them into a single measure, strange things occur near zero. Separating them out may be a way out.

  3. Thanks to everyone for responding.

    a. How important are the signs to you? For example, if a given company changes from -$1 to $1, is that more important to you than a change from $1 to $3, or from $3 to $5?
    They are the same importance. In the case of ties a change in sign might be more interesting, but in general, the -1 to 1 is the same to us as the 1 to 3. That's our gut instinct anyway, but we were hoping that someone could help us justify that approach. Currently it seems reasonable but ad hoc.
    b. For any given company, do you want to use the same scaling for all three years? I imagine the answer is Yes (so you don't have to worry about funny things happening such as an increase of 25%, followed by a decrease of 25%, does not bring things to the initial value).
    The scaling is the same, but we're mostly interested in the 3yr change:
    that is (2006-2004)/2004

    We came up with a few ideas for avoiding the problem, but none perfect. One is that we're using median and not mean as the sample statistic because that allows us to avoid the explosion around zero as long as it is far from the median. Since in our data this is the case, we can just replace zero with .001 and the median happily ignores it.

    The second was to look at the different groups and rather than measure magnitude, to just indicate the percentage improving. This doesn't seem to be quite as interesting a statistic, and discarding magnitude data seems not just wasteful but also ignoring something of critical interest.

    Marc, I'm confused by your suggestion, would you consider explaining it in greater detail.

    David, your approach seems interesting, but is adding one the right value? We add one to interest rates because rates are the return on a single dollar, whereas EPS are similar to yields but they are more like gross coupons on bonds than rates. What do you think about the following:
    Calculate an implied yield by taking 2003 EPS / 2003 share price. Gross return rate is then 1 + this dividend. Then you look at the same gross rate in 2006 and look at the percent change in that?
    Thanks,
    Benjamin Kay

  4. I vote for the rescaling approach – EBITDA (Earnings Before Interest, Taxes, Depreciation and Amortization) will not eliminate the problem (some companies have negative EBITDA, though that condition doesn't last long), but it should substantially reduce the problem and I believe the measure should be available through most sources that would also give EPS data. Best of luck!

  5. Benjamin — I think we're talking about the same thing: (earnings per share) / (price per share) = earnings/price.
    — David

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