# “Apple confronts the law of large numbers” . . . huh?

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

Measured by market capitalization, Apple 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.

P.S. Kaiser wrote about this nearly two years ago (even including the Apple Computer example, amazingly enough).

## 19 thoughts on ““Apple confronts the law of large numbers” . . . huh?”

1. Perhaps it should be read as “not just Apple shareholders, but also millions of investors who did not know they were Apple shareholders”.

Also: a value (wealth) going up rather than price is not a zero-sum game.

2. A large part of stocks do not get traded. Prices of stocks tend to incorporate future expected earnings. The zero-sum analogy goes on somewhere at the margin, with buyers paying more to sellers. But the all the stocks can count on higher expected future earnings.

3. (I.e. the price is a measure of value, but a price change in itself is a zero-sum phenomenon, at least to a first approximation.)

4. It seems that Warren Buffett agrees with your skepticism on the beneficence of stock price increases.

From page 7 of the 2011 shareholder letter:

“The logic is simple: If you are going to be a net buyer of stocks in the future, either directly with your own money or indirectly (through your ownership of a company that is repurchasing shares), you are hurt when stocks rise. You benefit when stocks swoon. Emotions, however, too often complicate the matter: Most people, including those who will be net buyers in the future, take comfort in seeing stock prices advance. These shareholders resemble a commuter who rejoices after the price of gas increases, simply because his tank contains a day’s supply.”

http://www.berkshirehathaway.com/letters/2011ltr.pdf

• Assuming the quote’s accurate, Buffet needs some help with logic and/or exposition.

The gas analogy is highly misleading, because gas is a consumable. You use it and it’s gone and you have to buy more.

After you buy a stock, you still have it. All else being equal, a market rising 10% per year is better than one rising 5% per year. Both are better than going down 5% per year in the long run.

Nobody wants to be a net buyer of stocks in the long run. That’d mean you’re losing money!

At most, Buffet means “buy low, sell high”. If you’re about to buy stock you think is undervalued (as Buffet professes to do), it’s even better if it goes down even further before you buy it. Then when it goes back up, you make more money.

5. There are several problems here, none of them related to Bernoulli.

The stock market as a whole can’t really increase faster than the rate of growth of wealth for very long (with due care given to defining the market as a whole to include foreign assets and privately held assets). So as Apple grows faster than other things, it comes to take a larger share of total wealth and is inevitably constrained in future growth, unless productivity as a whole behaves more like Apple than like everything else — and a longterm growth rate exceeding three percent per year is pretty much nirvana. looked at another way, the larger share Apple is of any particular index, the more they must outperform the average of the index in order to give a meaningsul difference in appreciation. If Apple is half the market, then outperforming the market by 10 percent will only look like an outperformance by 5 percent… because they’re half the index.

The second problem (owing nothing again to Bernuilli) is that it’s much much easier to grow a \$10 million company at 10 percent per year than it is to grow a \$100 billion company at ten percent per year. Your moves are now too big to be ignored by the market and engender counter-strategies. This is true bith financially and in the real world. That’s one reason why large mutual funds have lower median growth than small mutual funds (though smaller ones also have much higher volatilities and much larger failure rates.)

The fact that every stock sale has both a buyer and seller means that, prospectively, someone gained and someone lost — we just don’t know who, yet.

6. Excellent piece Andy. As for whether the rising stock price is of benefit, consider Warren Buffett who wakes up each day hoping prices fall for companies he owns – that way his reinvested dividends can pick up more shares, and the company gets a better deal on share repurchases.

BTW, I’m an old classmate of yours from Springbrook.

7. I lol’d.
Namespace pollution is the reason for this. In ‘business’ and ‘investment’ circles, the ‘Law of Large Numbers’ refers to the fact that it’s more difficult to keep up a growth rate from a large base. IE when Apple was small, it was easy to get 10% (or whatever) growth rates, but not they’re much larger, it’s more difficult.

http://www.investopedia.com/terms/l/lawoflargenumbers.asp

I can’t believe he used the term, and then looked up the wrong definition. Nice catch, and too funny.

8. The “price” of a stock is that of the last share bought and sold. The “price” is only the price for the marginal investor, and extrapolation of the value from about .00001% of the population to the entire population may be difficult.

Rising stock prices are probably a positive sum game simply because asset price inflation eases other constraints (credit, liquidity) which have real effects.

• Flatly flattered by your comment, EnlightenedDuck.

I’m nowhere close to Andrew’s level. However, I guess educated statisticians usually see things similarly 95 times out of 100.

9. I took the point to be that a large company’s revenues are the sum of many components. To the extent that the returns to those components aren’t perfectly correlated, some form of the LLN applies and we can expect the total return to revert to the mean.

Regarding your “unrelated note”: if half of Americans are share owners, aren’t they better off if their assets are worth more? It’s true that they can’t monetize their shareholding without selling to someone else, but so what? Presumably their share holdings are more valuable because the expected future return on the underlying assets has gone up, so the buyer is happy to pay more for the shares. (This is of course partly contradicting the assumption of the first point, that the mean return on assets is more or less fixed. But there’s some middle ground that makes both points reasonable.)

• You are confusing profit and revenue.

10. Stock not a zero-sum game. If firms are worth more in aggregate, who bought and who sold is more a question of how the gains are distributed. The general issue is that economic activity is typically a positive sum game, what some people call win-win. It’s a useful lesson, that life need not be like football or baseball.

11. I’ve long thought that journalists should be compelled to pay some reasonable forfeit – perhaps a single joint from one finger – for each time they misused ‘quantum’ or ‘exponential’. I imagine a few other words could be added to the list.

12. I think what Stewart is implicitly getting at is something like the following process:

Apply starts with a pot of investor’s money. It invests it, and get a positive amount back under some stochastic process. Something like lognormal centered at the original amount invested Consider however that each dollar has a draw associated with it. And after each period of time, the money is rolled over and invested again in the next period.

Thus it is more probable that a firm can get lucky in the beginning of the process, as less draws are made, so a very high variance can translate to occasional huge payoffs. But as the returns gets larger, so does the number of draws, so after a while we get asymptotic reversion to the mean.

This can sort of make sense if we think there’s size limit to any investment that Apple can make. So Apple ends up with a few investment projects in the beginning of its life, but simply has to diversify once it gets large enough since there even the latest Iphone doesn’t need \$100 billion dollars for development.