### Are CEOs overpaid or underpaid?

Corporate executives make a lot of money. Are they worth it? Are higher-paid CEOs actually better than their lower-paid counterparts?

Business Week magazine says, no, they're not, and they have evidence to prove it. They took 200 highly-paid CEOs, and did a regression to predict their company's stock performance from their chief executive's pay. The plot looks highly random, with an r-squared of 0.01. Here's a stolen copy:

The magazine says,

"The comparison makes it look as if there is zero relationship between pay and performance ... The trend line shows that a CEO’s income ranking is only 1 percent based on the company’s stock return. That means that 99 percent of the ranking has nothing to do with performance at all. ...

"If 'pay for performance' was really a factor in compensating this group of CEOs, we’d see compensation and stock performance moving in tandem. The points on the chart would be arranged in a straight, diagonal line."

I think there are several reasons why that might not be right.

First, you can't go by the apparent size of the r-squared. There are a lot of factors involved in stock performance, and it's actually not unreasonable to think that the CEO would only be 1 percent of the total picture.

Second, an r-squared of 0.01 implies a correlation of 0.1. That's actually quite large. I bet if you ran a correlation of baseball salaries to one-week team performance, the r-squared would probably be just as small -- but that wouldn't mean players aren't paid by performance. As I've written before, you have to look at the regression equation, because even the smallest correlation could imply a large effect.

Third, the study appears to be based on a dataset created by Equilar, a consulting firm that advises on executive pay. But Equilar's study was limited to the 200 best-paid CEOs, and that artificially reduces the correlation.

If you take only the 30 best-paid baseball players, and look at this year's performance on the field, the correlation will be only moderate. But if you add in the rest of the players, and minor-leaguers too, the correlation will be much higher.

(If you don't believe me: find any scatterplot that shows a strong correlation. Take a piece of paper and cover up the leftmost 90% of the datapoints. The 10% that remain will look much more random.)

Fourth, the observed correlation is fairly statistically significant, at p=0.08 (one-tailed -- calculate it here). That could be just random chance, but, on its face, 0.08 does suggest there's a good chance there's something real going on. On the other hand, the result probably comes out "too" significant because the 200 datapoints aren't really indpendent. It could be the case, for instance, that CEOs tend to get paid more in the oil industry, and, coincidentally, oil stocks happen to have done well recently.

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BTW, I don't think there's a full article accompanying the Business Week chart; I think what's in that link is all we get. Which is annoying, because it doesn't tell us how the 200 CEOs were chosen, or what years' stock performance was looked at. I'm not even sure that the salaries were negotiated in advance. If they weren't, of course, the result is meaningless, because it could just be that successful companies rewarded their executives after the fact.

Furthermore, the chart doesn't match the text. The reporters say they got an r-squared of 0.01. I measured the slope of the regression line in the chart, by counting pixels, and it appears to be around 0.06. But an r of 0.06 implies an r-squared of 0.0036, which is far short of the 0.01 figure. Maybe the authors rounded up, for effect?

It could be that my pixel count was off. If you raise the slope from 0.06 to 0.071, you now get an r-squared of 0.0051, which does round to 0.01. So, for purposes of this post, I'm going to assume the r is actually 0.07.

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A correlation of 0.07 means that, to predict a company's performance ranking, you have to regress its CEO pay ranking 93% towards the mean. (This works out because the X and Y variables have the same SD, both consisting of numbers from 1 to 200.)

In other words, 7 percent of the differences are real. That doesn't sound like much, but it's actually pretty big.

Suppose you're the 20th ranked CEO in salary. What does that say about your company's likely performance? It means you have to regress it 93% of the way back to 100.5. That takes you to 95th.

So, CEOs that get paid 20th out of 200 improve their company's stock price by 5 rankings more than CEOs who get paid 100.5th out of 200.

How big is five rankings?

I found a website that allowed me to rank all the stocks in the S&P 500 by one-year return. (They use one year back from today, so, your numbers may be different by the time you try it. Click on the heading "1-Year Percent.")

The top stock, Micron Technology, gained 151.47%. The bottom stock, Avon, lost 42.80%.

The difference between #1 and #500 is 194.27 percentage points. Divide that by 499, and the average one-spot-in-the-rankings difference is 0.39 percentage points.

Micron is actually a big outlier -- it's about 33 points higher than #2 (Facebook), and 52 points higher than #5 (Under Armour). So, I'm going to arbitrarily reduce the difference from 0.39 to 0.3, just to be conservative.

On that basis, five rankings is the equivalent of 1.5 percentage points in performance.

How much money is that, in real-life terms, for a stock to overperform by 1.5 points?

On the S&P 500, the average company has a market capitalization (that is, the total value of all outstanding stock) of 28 billion (.pdf). For the average company, then, 1.5 points works out to $420 million in added value.

If you want to use the median rather than the mean, it's $13.4 billion and $200 million, respectively.

Either way, it's a lot more than the difference in CEO compensation.

From the Business Week chart, the top CEO made about $142 million. The 200th CEO made around $12.5 million. The difference is $130 million over 199 rankings, or $650K per ranking. (The top four CEOs are outliers. If you remove them, the spread drops by half. But I'll leave them in anyway.)

Moving up the 80 rankings in our hypothetical example is worth only a $52 million raise -- much less than the apparent value added:

**Pay difference: $52 million**

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**Median value added: $200 million**

**Mean value added: $420 million**

Moreover ... the value of a good CEO is much higher, obviously, for a bigger company. The ten biggest companies on the S&P 500 have a market cap of at least $200 billion each. For a company of that size, the equivalently "good" CEO -- the one paid 20th out of 200 -- is worth three billion dollars. That's *60 times* the average executive salary.

Assuming my arithmetic is OK, and I didn't drop a zero somewhere.

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So, I think the Business Week regression shows the opposite of what they believe it shows. Taking the data at face value, you'd have to conclude that executives are

**underpaid**according to their talent, not

**overpaid**.

I'm not willing to go that far. There's a lot of randomness involved, and, as I suggested before, other possible explanations for the positive correlation. But, if you DO want to take the chart as evidence of anything,it's evidence that there is, indeed, a substantial connection between pay and performance. The r-squared of less than 0.01 only *looks* tiny.

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Although I think this is weak evidence that CEOs *do* make a difference that's bigger than their salary, the numbers certainly suggest that they *can* make that big an impact.

Suppose you own shares of Apple, and they're looking for a new CEO. A "superstar" candidate comes along. He wants twice as much money as normal. As a shareholder, do you want the company to pay it?

It depends what you expect his (or her) production to be. What kind of difference do you think a good CEO will make in the company's performance?

Suppose that, next year, you think Apple will earn $6.50 a share with a "replacement level" CEO. How much more do you expect with the superstar CEO?

If you think he or she can make a 1% difference, that's an extra 6.5 cents per share. That might be too high. How about one cent a share, from $6.51 instead of $6.50? Does that seem reasonable?

Apple trades at around 15 times annual earnings. So, one cent in earnings means about 15 cents on the stock price. With six billion Apple shares outstanding, 15 cents a share gives the superstar CEO a "value above replacement" of $900 million.

So, for a company as big as Apple, if you *do* think a CEO can make a 1-part-in-650 difference in earnings, even the top CEO salary of $142 million looks cheap.

Apple has the largest market cap of all 500 companies in the index, at about 15 times the average, so it's perhaps a special case. But it shows that CEOs can certainly create, or destroy, a lot more value than than their salaries.

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So can you conclude that corporate executives are underpaid? Not unless you can provide good evidence that a particular CEO really is that much better than the alternatives.

There's a lot of luck involved in how a company's business goes -- it depends on the CEO's decisions, sure, but also on the overall market, and the actions of competitors, and advances in technology in general, and world events, and Fed policy, and random fads, and a million other things. It's probably very hard to figure the best CEOs, even based on a whole career. I bet it's as hard, as, say, figuring baseball's best hitters based on only a week's worth of AB.

Or maybe not. Steve Jobs was fired as Apple's CEO, then, famously, returned to the struggling company a few years later to mastermind the iPod, iPhone, and iPad. Apple is now worth around 100 times as much as it was before Jobs came back. That's an increase in value of somewhere around $500 billion. It was maybe closer to $300 billion at the time of Jobs' death in 2011.

How much of that is due to Jobs' actual "talent" as CEO? Was he just lucky that his ethos of "insanely great" wound up leading to the iPhone? Maybe Jobs just happened to win the lottery, in that he had the right engineers and creative people to create exactly the right product for the right time?

It's obvious that Apple created hundreds of billions of dollars worth of value during Jobs' tenure, but I have no idea how much of that is actually due to Jobs himself. Well, I shouldn't say *no* idea. From what I've read and seen, I'd be willing to bet that he's at least, say, 1 percent responsible.

One percent of $300 billion is $3 billion. Divide that by 14 years, and it's more than $200 million per year.

If you give Steve Jobs even just one percent of the credit for Apple's renaissance, he was still worth 50 percent more than today's highest-paid CEO, 300 percent more than today's eighth-highest paid CEO, and 1500 percent more than today's 200th-highest-paid CEO.

Labels: Business Week, CEO, merit, r-squared, regression