# Apple Confronts the Law of Large Numbers?

I was reading this news article by famed business reporter James Stewart:

Measured by market capitalization, Apple (AAPL) is the world’s biggest public company. . . . Sales for the quarter that ended Dec. 31 . . . totaled \$46.33 billion, up 73 percent from the year before. Earnings more than doubled. . . . Here is the rub: Apple is so big, it’s running up against the law of large numbers.

Huh? At this point I sat up, curious. Stewart continued:

Also known as the golden theorem, with a proof attributed to the 17th-century Swiss mathematician Jacob Bernoulli, the law states that a variable will revert to a mean over a large sample of results. In the case of the largest companies, it suggests that high earnings growth and a rapid rise in share price will slow as those companies grow ever larger.

If Apple’s share price grew even 20 percent a year for the next decade, which is far below its current blistering pace, its \$500 billion market capitalization would be more than \$3 trillion by 2022. That is bigger than the 2011 gross domestic product of France or Brazil.

Later he writes, “Other companies that have reached the top appear to have been felled by Bernoulli’s law.”

It’s good to see some probability theory in the newspaper, even if the details got garbled in translation.

To be specific: the law of large numbers is a statement about the long-run average of a random process. I don’t think it really has anything to do with the mathematical property that exponential growth can’t go on forever. OK, I guess I see the analogy. If you think of a corporation’s finances as something like repeated plays at the roulette table, then Apple’s success can be though of as a long series of successful bets. But the success can’t go on forever; in the long run the corporation’s performance will revert to the mean. I don’t see how Bernoulli’s theorem adds to this, though.

This is no big deal; I guess my reaction is similar to the way physicists must feel when they see terms like “quantum jump” and “uncertainty principle” casually thrown around. Thus, please don’t consider this blog post as a rant; rather, it is a note on the challenges of mapping mathematical ideas to real-world applications.

Also, on an unrelated note, I was confused by this bit from Stewart’s article:

Apple shares have surged 68 percent from their low point in June, and it’s not just Apple shareholders who have benefited. Apple is now such a large part of the S.& P. 500 and the Nasdaq 100 indexes that it has buoyed millions of investors who own shares of broad index funds and mutual funds. These investors account for an estimated half of the American population.

I’m not an economist so I’m probably missing something here . . . but is it really correct to count an increase in Apple’s share price as a benefit for half the American population? If every seller has a buyer, then a 68% increase in stock prices means that someone out there is paying 68% more than they would have in June. To put it another way, if you already own an index fund then, sure, it’s great if the price goes up. But if you’re buying an index fund, an increase in the stock price implies that you’re paying more for the same thing. A price going up seems like a zero-sum game. Again, though, I’m no economist so maybe I’m missing something important.

Affiliation: Columbia University

Andrew Gelman is a professor of statistics and political science and director of the Applied Statistics Center at Columbia University. He has received the Outstanding Statistical Application award from the American Statistical Association, the award for best article published in the American Political Science Review, and the Council of Presidents of Statistical Societies award for outstanding contributions by a person under the age of 40.

His books include Bayesian Data Analysis (with John Carlin, Hal Stern, and Don Rubin), Teaching Statistics: A Bag of Tricks (with Deb Nolan), Data Analysis Using Regression and Multilevel/Hierarchical Models (with Jennifer Hill), and, most recently, Red State, Blue State, Rich State, Poor State: Why Americans Vote the Way They Do (with David Park, Boris Shor, Joe Bafumi, and Jeronimo Cortina).

Andrew has done research on a wide range of topics, including: why it is rational to vote; why campaign polls are so variable when elections are so predictable; why redistricting is good for democracy; reversals of death sentences; police stops in New York City, the statistical challenges of estimating small effects; the probability that your vote will be decisive; seats and votes in Congress; social network structure; arsenic in Bangladesh; radon in your basement; toxicology; medical imaging; and methods in surveys, experimental design, statistical inference, computation, and graphics.

Visit: Andrew Gelman's Website

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