1. Jim Caserta - June 29, 2006 at 3:55 pm

   First, plot the data with CEO pay on the y-axis and profits on the x-axis. Anything above the line is “bad”, pay > profit and below is “good” pay First, plot the data with CEO pay on the y-axis and profits on the x-axis. Anything above the line is “bad”, pay > profit and below is “good” pay < profit. There is another case where both are negative, which will also give an inaccurate result, pay=-10%, profit=-1000% pay/profit = 1/100, which would look good, but would be terrible!

2. James Grimmelmann - June 29, 2006 at 4:09 pm

   The ratio of (% change in CEO pay) to (% change in profit) is not a meaningful quantity. To see this most vividly, realize that a company with flat year-over-year profits will produce a division by zero. A company with a tiny profit increase will generate a hugely positive ratio; a company with a tiny profit decrease will generate a hugely negative ratio. This small perturbation in profits causes your ratio to swing wildly between very large and very small.

   To a first approximation, what you are thinking of is the ratio of (pay in year 2 / pay in year 1) to (profits in year 2 / profits in year 1). This ratio is, as you want to say, greater than 1 when pay is increasing faster than profits, and less than 1 when pay is increasing less quickly than profits. This measure has problems, as well (e.g. it blows up when the profits turn into losses in either year) but I think it’s what you’re thinking of.

3. M Dowd - June 29, 2006 at 4:29 pm

   Perhaps your hypotheses that “any value greater than or equal to 1 is bad” and “any any value less than 1 is good” are flawed. I think you mean to say that there is or should be a postive linear correlation between change in pay (”PAY”) and change in profits (”PP”).

   If you graph this in two-dimensions (PAY v. PP), then it should be clearer. The ideal change in pay should be any point on the line (determined by, e.g., linear regression of the data points) and the question is how much does each individual PAY/PP deviate from the industry average. (I recall a recent (2006) NYT or Wash Post article that did the same or similar analysis.)

   A negative (or positive) PAY/PP ratio is not by itself going to be indicative of a “good” or “bad” value. For example, consider these situations. Company A has -30% PP. The options for PAY are +20%, +1%, -1%, and -50%. Which of these are good or bad?

   If your hypothesis is that PAY should be reflective of PP, then wouldn’t you think that -30% PAY is what the CEO deserves?

   With the +20% PAY option, you get a negative ratio. This is clearly a bad relationship because the CEO gets a raise even though the company had lower profits, correct?

   With the +1% PAY option, you again get a negative ratio. This appears to be an instance where the CEO is overpaid for an underperforming company? CEO keeps the same pay essentially for an underperforming company.

   But with the -1% PAY option, you “unexpectedly” get a positive ratio, but again isn’t this an instance where the CEO is overpaid for an underperforming company? CEO keeps the same pay essentially for an underperforming company.

   With the -50% PAY option, you again get a positive ratio. Is this a good or bad relationship? Here the CEO, in my opinion, is being overly punished for the poor PP of -30%. This conclusion is different than what you and the other poster suggests.

4. Paul Gowder - June 29, 2006 at 5:36 pm

   Um, well, are you trying for real empirical results, or a normative conclusion about the behavior of a given corporation?

   If you’re trying for real empirical results, you would need some stat mojo to make meaningful generalizations about the relationship between CEO pay and profits, but presumably you know that.

   In terms of the normative conclusions about individual corporations, well, I *did* skip 12th grade calc, and pretty much learned this stuff on the street (don’t ask), so what I say following has no credibility whatsoever. (I mean, if you had functions for the pay and profit, it would be easy — take a couple of derivatives, whiz, bang…) Nonetheless…

   It seems like a partial solution to accounting for losses would be to do what James said, but take the absolute value of the resulting fraction. To the extent that profits and losses are within a narrow range, that should give you a reasonable evil > 1 > good map. For example, if year 1 profits and pay were both 15, and year 2 profits were -10 and pay 20, the absolute value of James’s ratio would be 2.

   Of course, if year 2 profits were -1000, that would break right away, because even the absolute value of the resulting ratio would be 1/500. But you might be able to fix that if profits and pay were both in a narrow range by mucking with the units — for example, always count the pay in billions and the profit in thousands.

   This is just off the cuff thoughts. I’ll give this some more thought later and play with some math, when I don’t have more productive things to do. If only because I find this sort of problem immensely interesting.

5. Vic Fleischer - June 29, 2006 at 7:02 pm

   Hi Elizabeth — why profits and not share price? CEOs with pay packages that were closely tied to profits (and not long term share price) might accelerate profits by, for example, underinvesting in advertising or R&D. But that might not be a good thing for shareholders or anyone else.

   Anyway, I’d start with Rob Daines’ paper (called something like CEO Pay – the Good, the Bad and the Lucky) and work from there. Vic

6. Tax Lawyer - June 29, 2006 at 7:04 pm

   First, I second Paul Gowder–you say you’re looking for a “relationship” between pay and profits–if you’re asking whether the two are correlated, you’re going to have to run a regression analysis. (Kate Litvak will certainly pop up soon to make an incredibly scathing yet incredibly true comment about this, and we’ll all be both sorry and glad she did.)

   Second, if you’re not going to run a regression analysis, there’s no reason to smush the numbers together. Is your normative point that a big pay raise shouldn’t accompany crappy profits? Okay, show the reader a chart and just say that.

   Third, if you want to smush the numbers together, may I suggest fractions of fractions, instead of positives and negatives? That is, if profits in year 2 moved -30% from year 1, and pay moved +50%, and your base is year one– that is, we’re interested in what happens to our pay/profit ration starting in year 1, so we normalize pay/profit in year one to equal 1–then we now have 1.5(pay)/.7(profits). This is bigger than one. Bad company! Bad CEO! But what if the CEO’s pay goes down 20% and the profits go up 10%? .8(pay)/1.1(profits) is less than one. Good CEO! This might be more what you’re looking for.

   If the company actually runs a net loss things will look a little different. This model doesn’t address that situation, and I would need to think more about it.

7. John Armstrong - June 29, 2006 at 7:05 pm

   Finally a question asking for a mathematician.

   I take it you want to generally say “percent increase for CEO pay greater than percent increase for profits bad”. If you plot the numbers with the CEO increase along the x-axis and the profit increase along the y axis, you want to label everything under the diagonal x=y as bad, and everything above it as good. The basic tool, then, is y-x. If it’s negative: bad. If it’s positive: good.

   However, this tends to lose information about how much bigger the CEO percent increase is than the profit percent increase. You want (for instance) all of the times the CEO is making twice the (percent) increase of the profits to be on a single curve x=2y. In general, such a curve will be a ray from the origin in some direction, and the closer to the diagonal the better (below) or worse (above). An idea might be to use the angle between the diagonal and the ray through the point, or more specifically the sine of that angle. If x=y, then this measure will give 0, while positive and negative values correspond to good and bad results. For your convenience, the formula is:

   (y-x)/sqrt(2(x^2+y^2))

   There’s one problem with this, though: it behaves oddly in the regions you’re already having trouble with. In particular, it would indicate that a CEO doubling his pay while the profits get cut in half is worse than doubling his pay while profits are quartered. I think I have an idea how to fix this, though, but I’ll have to ponder for a while and get back to you.

   Oh, and if anything I’ve said went over your head, feel free to email, though I’ll be driving tonight and won’t be able to check it between about 20:00 and 02:00.

8. Tax Lawyer - June 29, 2006 at 7:07 pm

   Oy. My comment is essentially James Grimmelman’s comment, but not as elegant. Disregard me. (Except I stand by the point about Kate Litvak.)

9. Paul Gowder - June 29, 2006 at 7:09 pm

   One further thought. My original suggestion above re absolute values doesn’t work because of paycuts. However, there’s an easy way to cut through all the trouble I think. Just forget about ratios. Subtract the percentage increase in pay from the percentage increase in salary. You’ll get a number that’s bad if it’s positive, and more bad if it’s more positive (doubling salary with 20% decrease in losses would give you 120, for example), good if it’s zero or negative. Then you can avoid the problem altogether, like magic.

   If you want to do some math on the other end that depends on getting a result around 1, you can always take the distribution of percentage differences and map them to appropriate fractions. I might work on the ratio thing more just for my own amusement, but I think dropping it altogether in favor of differences of percentages is much simpler and just as meaningful.

10. Elizabeth Nowicki - June 29, 2006 at 7:27 pm

    I need to rush out the door for a training run – I am preparing for the L&S Annual Meeting 5K (I imagine many of you are similarly preparing) – but I wanted to type out a THANK YOU before I go.

    Outstanding comments, all. Vic, I actually am looking at profit, eps, and stock price as my “measures of success.” Good minds think alike. Also, tax lawyer, I did do the basic charts, but I feel like a lousy analyst by just making charts and stopping there.

    I will come back here after my run and re-read everything. I will likely try to do all of the things y’all suggested. Thank you, again, for the feedback. John’s excellent formulaic suggestion makes me a bit… nervous, but I appreciate the feedback, and I will give it a whirl! Thanks.

    And, Tax Lawyer, I concur. No otherwise innocuous Nowicki thread is complete without a Kate Litvak drop-by. (Errrr, perhaps “drive by” is the better phrase.)

11. John Armstrong - June 30, 2006 at 4:19 am

    Okay, after much contemplation (punctuated by screaming down the vengeance of the gods on the entire Maryland Department of Transportation), I think I see a more fundamental flaw which hamstrings this whole effort from the beginning: percent increase is the wrong way to look at it.

    Consider this: is a 90% drop in pay (dividing by ten) more comparable in magnitude to a 90% increase (not quite doubling) or a 900% increase (multiplying by 10)? I’d say it’s the latter, and that’s the jumping-off point.

    So, instead of taking percent change, consider the multiplier from year to year. If you write the percent change as a decimal, you only have to add 1 to it to get the multiplier. Further, I’ll assume (please correct me if this is radically mistaken) that the multiplier is never negative — CEOs never have to pay the company to keep working, and companies never go from profit to loss. For CEOs this seems pretty safe, but I’d have to consider even further for the profit/loss assumption.

    Now to make good on this notion that a multiplier of 1/10 should be similar in impact to one of 10, we take the logarithm (don’t faint, now, I’ll try to be gentle) to get the ‘exponent’. At first blush, the base doesn’t matter, so I’ll take it to be 10 for now (corresponding to the ‘log’ button on your calculator). Then a multiplier of 1/10 corresponds to an exponent of -1, while 10 corresponds to an exponent of 1. The absolute values are the same, which is how they can be compared. In the sequel I’ll use the function f defined as

    Now, if the percent increases are the same, then the multipliers are too (just add 1), and thus the exponents are too (log of each). If the percent increase for the CEO’s pay (which I’ll call c) is greater than that for profit (called p), then the CEO exponent is greater than the profit exponent. So we can still tell whether the CEO is outdoing the company.

    So how can we tell how much better? I propose the comparison function

    log(1+p) – log(1+c) = log((1+p)/(1+c))

    This increases as p increases while c stays the same (bigger profit percent increase for same pay increase is better), and decreases as c increases while p stays the same (bigger pay increase for same profit increase is worse).

    Actually, if calculating a logarithm is big and scary (either for the author or her audience), you can just get by with (1+p)/(1+c) instead of your originally proposed p/c, and things should still work. The benefit of the logarithmic comparator is that given two data points with the same absolute value but opposite signs, the one is as “good” as the other is “bad”. The simpler version makes this comparison of magnitudes less clear to a semi-numerate reader.

    Also, with the logarithmic comparator it’s a lot easier to change the norm: if a 1% raise for maintaining profits (0 profit change) is to be considered the norm, then add 0.01 to the comparator to give that (hypothetical) data point a value of (almost[*]) zero. If a CEO should raise profits by 1% to justify keeping the same amount of pay, subtract 0.01.

    I hope that this method (a) is easy enough to implement, (b) makes sense to you, and (c) helps make whatever point you’re trying to make. As before, feel free to email if you have any more questions about (this or other) math. It’s nice to feel useful.

    [*] Please note that though 1% = 0.01, the real value you should add or subtract is log(1.01) = 0.00995… It just turns out that for small enough x, log(1+x) is almost exactly x.

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