# Apple and the Law of Large Numbers

This is a good but flawed New York Times piece which reflects on Apple’s staggering growth. This week, Apple stock hit a record high of \$526 per share. The bulk of the piece focuses on the so-called Law of Large Numbers in assessing Apple’s growth:

Apple is so big, it’s running up against the law of large numbers.

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.

Unfortunately, the writer of the piece (and its editors) don’t fully grasp the meaning of “Law of Large Numbers.” That law states that if you perform an experiment enough times, the average of results will approximate the expected value of the random variable. Here’s the rub: you can use this law to predict the behavior of experiments where you can deduce (or solve for) the expected value. For instance, if you toss a fair coin enough times, the Law of Large Numbers implies that the coin will land on heads (approximately) equal number of times as tails. Similarly, if you toss a fair six-sided die enough times, and look at the face value, the result should approach the expected value of 3.5 (the average of 1 + 2 + 3 + 4 + 5 + 6).  But you can’t use the Law of Large Numbers for experiments where you can’t deduce the expected value. Who’s to say that Apple’s earnings or share price should follow a certain reversion to the mean? What if we’re witnessing a novel company that is going to break all kinds of records? I think this is the case here.

The New York Times piece then attempts to justify the Law of Large Numbers by citing examples such as the fall of Cisco systems from a record \$557 billion market capitalization to close to \$100 billion today. Again, the major assumption there is that stocks tend to behave in a similar fashion, and that history repeats itself.

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Disclosure: I am long AAPL.