What is Apple Worth? Part 1: Regarding Large Numbers

A Big Lottery

“Peter tosses a coin and continues to do so until it should land "heads" when it comes to the ground. He agrees to give Paul one ducat if he gets "heads" on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled. Suppose we seek to determine the value of Paul's expectation.”

This is how Daniel Bernoulli rephrased a problem nagging a tight circle of mathematicians almost 300 years ago. The issue is that, if you go by the mathematical expected value of the winnings you end up with an infinite value for the lottery: you have 1/2 probability of winning one ducat on the first throw, 1/4 probability of winning two ducats in 2 throws (which adds another 1/2 to the expected value), 1/8 probability of winning 4 ducats in 3 throws (adding another 1/2), and so on adding 1/2 a ducat in value an infinite number of times.

The expected value suggests that a player should be willing to pay any amount asked for a chance to play this lottery. Yet, in real life no one in their right sense would pay more than a few ducats to play this, perhaps somewhere around 5 ducats. Why is it that the mathematical expected value doesn’t reflect the real value placed by real players on this lottery?

Mobile Apple: The Limits to Growth (Part 3)

Even the largest avalanche is triggered by small things.

Vernor Vinge

It is better to err on the side of daring than the side of caution.

Alvin Toffler

(here's Part 1 and Part 2 of this series)

Mac: A Recap

In the previous part of this series, I left with a chart of the Mac unit sales over the last few years, contrasting those units sold into the existing installed base as replacement Macs, and those representing new users adopting the Mac platform. I'll begin this post with a similar chart, this time spanning a few more years back, and smoothing out the quarterly data through the use of a trailing 4-quarters window to compute the values charted. Also, I've included a couple years worth of estimates into the future (period denoted in gray) after a reader's request. Check it out (click to enlarge).

It's clear from this chart that the slow 5-year life cycle of traditional PCs, combined with the practically unlimited potential for expansion into the billion-and-a-half units within the overall PC installed base, precludes us from modeling any significant slowdown in the rate of adoption for Macs within the next few years. Perhaps only after a significant slowdown in wider PC adoption (currently at over 100m new users per year), and when Mac adoption surpasses 50% share of those new-to-PCs users (currently around 11%), one might consider an inflection in the incremental adoption of Macs (explore the charts on the previous posts to contextualize these two opposing trends).

Mac: The Limits to Growth (Part 2)

To give full growth to that which still doth grow?

(this post is a continuation of this one)

If you haven't read the first part of this exercise, please at least skim through it so you can take better advantage of whatever insights might come from the analysis presented here.

I'm going to try to observe the evolution of the size of customer installed bases, which should gradually expand (or contract) in typical s-shaped patterns (aka sigmoid, logistic or other similar functions) when plotted over time. This derives from the notion that significant, long-term changes in customer adoption driven by innovative technologies require time to play out, as different segments of customers react differently to change, even assuming immediate information transfer about the innovation (which is not the case even in the internet age).

I'll tend to focus the charts around easily visualizing the yearly variation of the customer bases considered (y' continuously but ∆y given our discrete data) which should roughly fit a bell shaped curve. The shapes don't need to be exactly as the figures shown previously since those were just two very simplified examples given arbitrary theoretical assumptions on how the relative growth rate y/y should moderate as y gets bigger (notice the subtle distinction between y'/y=relative rate of change refers to the continuous modeling notation and "y/y=year over year"or "q/q=quarter over quarter" as the discrete modeling shorthand for the same measure).

So, for this second part I'll start with a broad historic view of all computing, and later dig into the adoption of traditional personal computers highlighting Apple's participation of such adoption, while aiming to give some perspective on the Mac's potential opportunity for expansion within that market. I'll leave a more detailed scrutiny of the recent mobile revolution for a later post.

Apple: The Limits to Growth (Part 1)

‘Rule Forty-two. All persons more than a mile high to leave the court .’

Everybody looked at Alice.

"Population, when unchecked, increases in a geometrical ratio, Subsistence, increases only in an arithmetical ratio."

–Thomas Malthus

"Evolution is the result of a sequence of replacements."

–Elliot W. Montroll

Over the last 8 years Apple's revenue has grown from $6.6b in calendar 2003 to $128b in calendar 2011, a 19.4x increase, or an annualized compound rate of 45% per year. Most astonishingly, EPS (including some substantial share dilution) has grown from 13 cents in 2003 to a whopping $35.11 in 2011, a 270x increase, or annualized to a compound rate of 101% per year. That's more than the equivalent of doubling every single year for eight years (2^8=256 times).

In light of such an impressive long-term performance, last quarter's 73% and 116% Y/Y growth, in revenue and EPS respectively, may seem like par for the course. Indeed, this is the way many "savvy" yet uninformed investors characterize Apple and AAPL: as a momo growth stock bound to fall off a cliff (so many examples from last year: NFLX, RIMM, FSLR, APKT, SINA, just to name a few) as soon as the scorching growth cools down a bit (and it will sooner or later, that's an undistputable fact).