# Apple and the myth of "The Law of Large Numbers"

### As Apple continues to set new records for revenue and profits seemingly every single quarter, it's become common for analysts and Apple observers alike to say that Apple's tremendous growth can't continue because the company is quickly running into the law of large numbers. Here's why they're wrong.

As Apple continues to set new records for revenue and profits seemingly every single quarter, it's become common for analysts and Apple observers alike to say that Apple's tremendous growth can't continue because the company is quickly running into the law of large numbers.

The law of large numbers, in math, is a statistical model which holds that as the number of times an experiment is repeated (or sample population increases), the average of the results will revert back to the mean. For instance, as you steadily increase the number of coin flips, the average result will increasingly inch closer to a 50/50 distribution between heads and tails.

As applied to the financial realm, the law of large numbers has been morphed into something wholly different, and much less mathematically sound, at least as employed by analysts.

When people reference the law of large numbers with respect to Apple, what they really mean to say is that Apple can't keep growing at 20+% increments indefinitely because it will eventually become so large that maintaining these growth levels becomes practically impossible.

After all, a company with \$100 million in revenue need only earn an additional \$20 million to increase revenue by 20% while company with revenue of \$40 billion needs to earn additional \$8 billion to achieve the same 20% increase in revenue. Clearly, the latter is much more daunting.

Put simply, the law of large numbers states that as companies grow larger and larger, their rate of growth will inevitably slow down, and that's where some many analysts feel Apple is already at.

And looking at Apple's recent financials, it's easy to see why some believe Apple is poised for a slow down.

Here are some stats for you.

• Apple has delivered 20+% growth rates every single quarter except for one going all the way back to 2003
• Over the last 5 years, Apple's annual revenue has increased by a factor of 5.6
• In 2011, Apple earned more money (\$108 billion) than it did in 2009 and 2010 combined (\$101.77 billion)
• Apple's least profitable quarter in 2011 (\$ 24.67 billion in Q1) was greater than Apple's most profitable quarter in 2009 (\$15.68 billion)

Apple's tremendous growth reached unprecedented heights in the December quarter of 2011 when Apple recorded all-time records in revenue (\$46.33 billion), profits (\$13.06 billion) and EPS (\$13.87).

Anchoring these strong results were all-time quarterly sales records of iPhones, iPads, and Macs.

So clearly, the party has to end some time right?

I mean, there's just no way Apple can keep on delivering these types of results, right?

Maintaining impressive growth rates when revenue is in the tens of billions just can't be done,  and hey, shouldn't the law of large numbers be kicking in any quarter now?

After all, Apple's growth - as illustrated above - has been absolutely off the charts.

With growth rates in the double digits, analysts are skeptical that Apple can maintain this type of growth.

But as is typically the case, Apple is poised to buck this trend longer than most people realize.

The New York Time recently covered Apple and the law of large numbers and cited analyst Robert Cihra who said, “over the past couple of years, they have actually accelerated revenue growth. I don’t know that can continue indefinitely. If you extrapolate far enough out into the future, to sustain that growth Apple would have to sell an iPhone to every man, woman, child, animal and rock on the planet.”

The reality, however, is much more nuanced.

People like to look at Apple and ascertain how much more money Apple has to earn in order to maintain the impressive growth rates investors have enjoyed over the past few years. But in doing so, they completely miss the point and ignore the variables that actually matter.

When considering the law of large numbers, it makes much more sense to study a company's marketshare and prospects for growth than it is to look exclusively at their current revenue and past performance. Yet that's what analysts who like to tout the law of large numbers tend to do. They look at Apple's size and reflexively bring up the "law of large numbers" and shout, "Ya see how big Apple is?! There's no way it can continue its phenomenal growth, there's just no way!"

But the real operative variable isn't revenue, but rather the factors that drive revenue - margins and marketshare. And when it comes to those two metrics, Apple is sitting pretty.

Apple is an exceptionally unique company in that it has impressively high margins with relatively scant marketshare. In other words, it makes more with less.

Take the iPhone for example. The iPhone is Apple's main source of revenue but it still lags behind Android in terms of US marketshare. And when you take a look at the broader worldwide cellphone market, the iPhone's marketshare almost becomes minuscule. In April 2011, IDC pegged Apple's worldwide share of the mobile phone market at 5% while Gartner's estimate came in at 4.6%.

But when we look at smartphone profits, Apple is taking home most of the cash. As of February, estimates put Apple's profit share in the smartphone market as high as 75%. Meanwhile, it's revenue share checks in at 40%.

Back in February, Horace Dediu of Asymco illustrated Apple's appetite for profits with this telling chart.

Clearly, Apple's margins are the best in the business.

Apple does have one product with a lions share of marketshare - the iPad. But even in the tablet space where Apple, by some estimates, commands a 85+% marketshare, Apple's future looks bright. Because the tablet market is still in its relative infancy, and by some accounts set to explode over the next few years, Apple still has plenty of room for growth ahead. Many, including Apple CEO Tim Cook, believe that the tablet market is poised to explode and will eventually overtake PCs in unit sales.

During Apple's last earnings conference call, Cook said, "I think there will come a day that the tablet market is larger than the PC market.”

And then there's the Mac.

AppleInsider this past June highlighted Apple's healthy margins by estimating Apple's average profit per sale of one Mac vs HPs profit from the sale of one PC.

Again, Apple's margins here are quite healthy.

Over the past few years, Mac sales have increased quite steadily, though not quite as dramatically as the iPhone or iPad. Still, the gains are impressive, due in large part to the halo effect brought upon by the wild success of the iPod and iPhone.

Apple's Mac sales in the December quarter of 20xx:

• 2007 - 2.31 million units
• 2008 - 2.52 million units
• 2009 - 3.36 million units
• 2010 - 4.13 million units
• 2011 - 5.2 million units

And still, Apple's share of the PC market continues to hover in the 10% range at best.

So if we look at Apple's top 3 revenue drivers - the iPhone, the iPad, and the Mac - Apple isn't even close to hitting a ceiling any time soon.

Apple's almost comically high revenue isn't an indication that growth is poised to slow down, but instead reflects Apple's healthy margins and their ability to make more money with less marketshare.

A little iPhone accounting never hurt nobody

Further illustrating Apple's room for revenue and profit growth, basic managerial accounting shows that Apple doesn't necessarily need to double its iPhone sales in order to double its iPhone profits.

Here's a quick and dirty example showing how a company can sometimes double revenue while only increasing sales by 20%.

Apple's iPhone profits = (The profit derived from each iPhone sold x sales volume) - fixed costs

Fixed costs are business expenses associated with iPhone production that don't fluctuate based on increases in manufacturing throughput or units sold.  These costs might include anything from advertising costs to iPhone research/testing costs. Put simply, fixed costs remain constant no matter if Apple sells 100 iPhones or 100 million iPhones.

Keeping the math easy, let's assume each iPhone is sold nets Apple a \$15 profit and that Apple's fixed costs check in at \$1.2 million.

With sales volume of 100,000 units, Apple's iPhone profits = (\$15/unit x 100,000 units) - \$1.2 million = \$300,000.

Now let's assume Apple increases iPhone sales in the next quarter to 120,000 units, a 20% increase.

In that scenario, Apple's iPhone profits = (\$15 x 120,000) - \$1.2 million = \$600,000

As you can see from this hypothetical example, Apple was able to increase its iPhone profits by 100% even though iPhone sales only increased by 20%.

The point is that an increase in iPhone sales, for example, can easily yield a disproportionate increase in profits. This is easily applied to the iPad and the Mac as well.

The law of large numbers will only and truly begin to affect Apple once they amass significant marketshares across the entirety of their product line, and to be clear, this has nothing to do with mathematical probabilities.

And even in that scenario, Apple has proven to be quite adept at generating new revenue streams just when analysts think that they've topped out. For years, analysts have continuously said that Apple's tremendous annualized growth was bound to peter out, that there was no way for them to keep it up. "Apple's best days, growth wise, are behind it," they'd all say.

And yet, without fail, Apple has not only beat Wall St. earnings estimates nearly every quarter, but has absolutely demolished them.

When the iPod reigned supreme, analysts were quick to ask, "What's next?" And then came the iPhone. And after that came the iPad. And who knows, maybe Apple's next big product - perhaps the rumored HDTV - will be Apple's next big revenue stream.

The bottom line is that all of this talk regarding Apple's growth beginning to stall is laughably premature.

Whether Apple can continue to deliver outstanding quarterly earnings result is a valid question, but that analysis should focus on marketshare, margins, new products, new revenue streams, increased competition - all of these are much more relevant to an analysis concerning Apple's future financial prospects.

Just because Apple's financials are impressively high doesn't, in and of itself, mean they can't get higher.

As an illustrative anecdote, the Confounded by Confounding blog writes:

I’d also like to note that, years ago, when Mr. Dell was calling for Apple to be scrapped and the proceeds given to shareholders, and it was trading for very low double digits instead of middling-high triple digits, no one was going “You know, the Law of Large Numbers will eventually drag Apple up. It’s a sure thing!”

If Apple's growth begins to stagnate soon, it'll have nothing to do with the law of large numbers. It will be because they failed to innovate, increase marketshare, and drive sales.

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